List of Divisibility Rules

For explanation of general divisibility rules explained, see Divisibility Rules.

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Below is a List of Divisibility Rules sorted by number.

Difficulty Coding

Very Easy

Easy

Somewhat Easy

Medium

Moderately Hard

Difficult

Very Difficult

Totally Nuts

Additionally, if an alternative method is available (albeit it would be hard most of the time), the description for that number has a pink background.

Divisibility of Numbers below 5

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	1	Every integer is divisible by one. All prime numbers are divisible by one, too. (Proof)Proof for Divisibility of 1: There is no need for a proof here. Any number is divisible by one. This rule does not apply to differentiate prime from consecutive numbers.Every integer is divisible by one. All prime numbers are divisible by one, too. (Proof)Proof for Divisibility of 1: There is no need for a proof here. Any number is divisible by one. This rule does not apply to differentiate prime from consecutive numbers.	4,623 is divisible by one; 91,237 is divisible by one.4,623 is divisible by one; 91,237 is divisible by one.
	2	Every number including 0 which ends with 0, 2, 4, 6 or 8 is divisible by two. (A number is divisible by 2 only if its last digit is 0, 2, 4, 6 or 8.) (Proof)Proof for Divisibility of 2: We have a base-10 number system so that when a number gets to 8 and we add two, it carries over and the last digit is 0 (because 8+2=10). Now the cycle repeats (0+2=2, 2+2=4, 4+2=6, 6+2=8) and we continue carrying over forever.Every number including 0 which ends with 0, 2, 4, 6 or 8 is divisible by two. (A number is divisible by 2 only if its last digit is 0, 2, 4, 6 or 8.) (Proof)Proof for Divisibility of 2: We have a base-10 number system so that when a number gets to 8 and we add two, it carries over and the last digit is 0 (because 8+2=10). Now the cycle repeats (0+2=2, 2+2=4, 4+2=6, 6+2=8) and we continue carrying over forever.	65,156,151,594 is divisible by two because the units digit is 4, and 1,597,534,568,852 is divisible by two because the units digit is 2.65,156,151,594 is divisible by two because the units digit is 4, and 1,597,534,568,852 is divisible by two because the units digit is 2.
	3	A number is divisible by three if the sum of its digits is divisible by three. (Proof)Proof for Divisibility of 3: This method works for divisors that are factors of 10 − 1 = 9. First of all, 9 = 10 − 1, which means 10 ≡ 1 mod 3. Now, if we are able to raise everything to the nth power, then we get 10n≡ 1n≡1 (mod 3). Since two things that are congruent modulo 3 are either both divisible by 3 or both not, we can interchange values that are congruent modulo 3. So, in a number such as the following, we can replace all the powers of 10 by 1. Afterwards, the equation gives us: $100a + 10b + 1c \equiv 1a + 1b + 1c \pmod{3}$, which is exactly the sum of the digits.The advanced method is a lot less efficient. Trim the rightmost digit. Add the digit to the rest of the number.	Is 12,423 divisible by 3? (1) Add all the digits together, we get 1 + 2 + 4 + 2 + 3 = 12. (2) 12 is divisible by 3, so 12,423 is, too.Is 12423 divisible by 3? (1) Trim off the rightmost digit, 3. Add it to 1242. 1242 + 3 = 1245. (2) Repeat the steps: 124 + 5 = 129. 12 + 9 = 21. 2 + 1 = 3.
	4	A number is divisible by four if and only if its last two digits of the number are divisible by four. (Proof)Proof for Divisibility of 4: Since any number divisible by 4 that is divided by 4 has a remainder of 0, we can write the following, using that number as 100, let's say: $100 \equiv 0 (mod 4), 10_k \equiv 0 \pmod{4} for k=2, 3, 4 etc...$ $x \equiv a_0 + a_1 (10) + a_2 (0) + \ldots + a_k 0 \pmod{4}$ $\equiv a_0 + a_1 (10) \pmod{4}$ Therefore, x is divisible by 4 if and only if the number is divisible by 4 where a0 + a1 (10) mod 4 is the number formed by the last two digits of x.Alternatively, add 2 times the tens digit to the ones digit, then check the divisibility.	156,128 is divisible by four, and 6,416 is divisible by four. The last two digits are 28 and 16, respectively, both of which are divisible by 4.For 156,128, 2 x 2 + 8 = 12. 12 is divisible by 4. Therefore, 156,128 is divisible by 4.
	5	A number is divisible by five if and only if the last digit is a 0 or 5. (Proof)Proof for Divisibility of 5: We first assume that every number ending in 0 or 5  is not divisible by five. Take the equation: y=5x where x is any integer constant.  Therefore, the value 5x ends with 0 or 5, since 5 multiplied by an integer constant ends with 0 or 5.   Dividing both sides by five, we get y/5 = x, an integer constant.  Therefore, even if we take the limit when x approaches infinity, we still get an integer constant.  This contradicts the assumption that every number ending in 0 or 5 is not divisible by 5.A number is divisible by five if and only if the last digit is a 0 or 5. (Proof)Proof for Divisibility of 5: We first assume that every number ending in 0 or 5  is not divisible by five. Take the equation: y=5x where x is any integer constant.  Therefore, the value 5x ends with 0 or 5, since 5 multiplied by an integer constant ends with 0 or 5.   Dividing both sides by five, we get y/5 = x, an integer constant.  Therefore, even if we take the limit when x approaches infinity, we still get an integer constant.  This contradicts the assumption that every number ending in 0 or 5 is not divisible by 5.	213,478,765 is divisible by 5 because the last digit is 5.213,478,765 is divisible by 5 because the last digit is 5.

Divisibility of Numbers between 6 and 10

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	6	Because its prime factorization is 2 x 3, it should be divisible by those two numbers. (A number is divisible by 6 only if it is even and the sum of its digits is divisible by 3.)Because its prime factorization is 2 x 3, it should be divisible by those two numbers. (A number is divisible by 6 only if it is even and the sum of its digits is divisible by 3.)	Is 4,044 divisible by 6? 4,044 is even, so it is divisible by 2. 4 + 0 + 4 + 4 is 12, so it is divisible by 3. Therefore, 4,044 is indeed a multiple of 6.Is 4,044 divisible by 6? 4,044 is even, so it is divisible by 2. 4 + 0 + 4 + 4 is 12, so it is divisible by 3. Therefore, 4,044 is indeed a multiple of 6.
	7	Take the last digit of the number, multiply it by two, and then subtract it to the remaining digits of the number. Repeat the process until the result can be easily identified. (Proof)Proof for Divisibility of 7: Every number can be expressed in the form $10x+y$, where y is the units digit. At this stage, we are trying to prove that if $x-2y$ is divisible by 7, then $10x+y$ is also divisible by 7. If $x-2y$ is divisible by 7, we can write: $x-2y=7p$, where p is a natural number. Multiply both sides by 10: $10x-20y=70p$ Add 21y to both sides:$10x+y=70p+21y$ Since $70p+21y$ is divisible by 7, we have concluded that $10x+y$ is divisible by 7 if $x-2y$ is divisible by 7. Another strategy is to add 5 times the last digit to the remaining digits instead. If the number is too large for the previously mentioned method to be viable then a block strategy can be used. First solit the number in blocks of 3 digits each, then start by substrating the first block with the second, add the third block to the result, substract the fourth one and repeat the pattenr, alternating betweem addition and substraction, until you get one single number. If that result is divisible by 7 the original number is also divisible by 7.Sort the numbers in blocks of three from right to left. Then, add the first group from the right, subtract the second group, then add the third, subtract the fourth, and so on. This is called the alternating sum of three. If the result is the multiple of 7, then the number is divisible by 7.	Is 3,409 is divisible by 7? First, take 9 from 3409, and multiply 9 by 2 (9*2=18). Then, subtract the doubled digit to the remaining digits. (340-18=322) Repeat the process: take 2 from 322 and multiply it by 2 (2*2=4). Subtract the doubled digit to the remaining number (32-4=28) 28 is divisible by 7, so 3,409 is also divisible by 7. Is 1,702,906,247 divisible by 7? First, group the numbers into blocks of three: 247, 906, 702, 1. Then, form the alternating sum: 247 - 906 + 702 - 1 = 42 Examine the results. Since 42 = 7 * 6, 42 is divisible by 7, thus 1,702,906,247 is, too.
	8	It is very simple, check whether the last three digits of the number is divisible by 8, if it is, then the number is divisible by 8.If the hundreds digit is odd, add 4 to the last 2 digits, and then examine the number; if it is even, simply check the last 2 digits directly.	Is the number 7,377,473,496 divisible by 8? First, Take the last three digits of the number. The last three digit is 496, so check whether it is divisible by 8. 496 is divisible by 8. Therefore, 7,377,473,496 is divisible by 8.For the number 362,744, first check the hundreds digit. The hundreds digit is odd, therefore 4 is added to the last two digits to examine. 48 is divisible by 8; therefore, 362,744 is divisible by 8.
	9	Add all the digits together. Then, instead of checking divisibility of 3, check if it is the multiple of 9. (Proof)Proof for Divisibility of 9: 9 = 10 − 1, which means 10 ≡ 1 mod 9. Any number which is one less than an integer power of ten is divisible by 9. This can be proved by expressing each number in this group as a term in the sequence Tn=T(n-1)*9+9. If T(n-1) is divisible by 9, Tn will be also. The first term in this sequence is 0 (100-1) which is divisible. Therefore, if each digit of a number is expressed as its actual place value as a sum of terms where a is the first digit, b is the second, etc. it is congruent to the sum of the digits as shown: $1000a + 100b + 10c + 1d \equiv (1000-999)a + (100-99)b$ $+ (10-9)c + (1-0)d \equiv a+b+c+d \pmod{9}$ , which is the sum of the digits.Add all the digits together. Then, instead of checking divisibility of 3, check if it is the multiple of 9. (Proof)Proof for Divisibility of 9: 9 = 10 − 1, which means 10 ≡ 1 mod 9. Any number which is one less than an integer power of ten is divisible by 9. This can be proved by expressing each number in this group as a term in the sequence Tn=T(n-1)*9+9. If T(n-1) is divisible by 9, Tn will be also. The first term in this sequence is 0 (100-1) which is divisible. Therefore, if each digit of a number is expressed as its actual place value as a sum of terms where a is the first digit, b is the second, etc. it is congruent to the sum of the digits as shown: $1000a + 100b + 10c + 1d \equiv (1000-999)a + (100-99)b$ $+ (10-9)c + (1-0)d \equiv a+b+c+d \pmod{9}$ , which is the sum of the digits.	Is 14625 divisible by 9? Add all the digits together: 1 + 4 + 6 + 2 + 5 = 18 18 is divisible by 9, so 14625 is, too.Is 14625 divisible by 9? Add all the digits together: 1 + 4 + 6 + 2 + 5 = 18 18 is divisible by 9, so 14625 is, too.
	10	If the last digit of the number is 0, it is divisible by 10.If the last digit of the number is 0, it is divisible by 10.	158,375,209,580 is divisible by 10 because the last digit is 0.158,375,209,580 is divisible by 10 because the last digit is 0.

Divisibility of Numbers between 11 and 20

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	11	If the difference of sum of digits at odd places and sum of digits at even places is 0 or divisible by 11, then the number is divisible by 11.If the difference of sum of digits at odd places and sum of digits at even places is 0 or divisible by 11, then the number is divisible by 11.	Is 527,901 divisible by 11? First, Add the digits at odd places. (5+7+0=12) Add the digits at even places. (2+9+1=12) Subtract both sums. (12-12=0) The result is 0, so 527,901 is divisible by 11.Is 527,901 divisible by 11? First, Add the digits at odd places. (5+7+0=12) Add the digits at even places. (2+9+1=12) Subtract both sums. (12-12=0) The result is 0, so 527,901 is divisible by 11.
	12	Because its prime factorization is 22 x 3, the number needs to be divisible by 4 and 3.Because its prime factorization is 22 x 3, the number needs to be divisible by 4 and 3.	9,180 ends in 80 (4*20), so it is divisible by 4. The sum of all digits is 18, so it is divisible by 3. Therefore, 9,180 is a multiple of 12.9,180 ends in 80 (4*20), so it is divisible by 4. The sum of all digits is 18, so it is divisible by 3. Therefore, 9,180 is a multiple of 12.
	13	Take the last digit of the number, multiply it by four, and then add it to the remaining digits of the numbers. Repeat the process until the result can be easily identified. Alternatively, subtracting 9 times the last digit to the remaining digits is also a viable strategy.Form the alternating sums of three. After sorting the numbers in blocks of three, add the first group from the right, subtract the second group, then add the third, subtract the fourth, and so on. At last, check if the result is the multiple of 13 (If needed, use the first method again to further reduce down a 3-digit number).	Take 85,969 for example. Take out 9 from 85969 and multiply it by four. (9 x 4 = 36) Add it to the remaining digits. (8596 + 36 = 8632) Take out 2 from 8632 and multiply it by four. (2 x 4 = 8) Add it to the remaining digits. (863 + 8 = 871) Take out 1 from 871 and multiply it by four. (1 x 4 = 4) Add it to the remaining digits. (87 + 4 = 91) 91 is divisible by 13, so 85,969 is also divisible by 13.Take 3,692,513,279 for example: Alternating sum of three: -3 + 692 - 513 + 279 = 455 After that, use the first method to determine if 455 is a multiple of 13. Take out 5 from 455 and multiply it by 4. (5 x 4 = 20) Add it to the remaining digits. (45 + 20 = 65)
	14	As 14 is the product of 2 and 7, perform divisibility checks of both 2 and 7.As 14 is the product of 2 and 7, perform divisibility checks of both 2 and 7.	For the number 745,514: It is divisible by 2, since it is an even number. Perform divisibility check of 7.Divisibility of 7: Take away the last digit of the number, multiply it by 2, and subtract it from the remaining digits. Repeat the process until the result can be determined. Example: For the number 745,514, 1. Take the last digit of the number and multiply it by 2. (4 x 2 = 8) 2. Subtract that number from the rest of the digits. (74551 - 8 = 74543) 3. Repeat the process. (7454 - 3 x 2 = 7448) 4. Continue until the results can be determined. (744 - 8 x 2 = 728) 5. The result is should be divisible by 7. (72 - 8 x 2 = 56) 56 is divisible by 7. Therefore, 745,514 is divisible by 14.For the number 745,514: It is divisible by 2, since it is an even number. Perform divisibility check of 7.Divisibility of 7: Take away the last digit of the number, multiply it by 2, and subtract it from the remaining digits. Repeat the process until the result can be determined. Example: For the number 745,514, 1. Take the last digit of the number and multiply it by 2. (4 x 2 = 8) 2. Subtract that number from the rest of the digits. (74551 - 8 = 74543) 3. Repeat the process. (7454 - 3 x 2 = 7448) 4. Continue until the results can be determined. (744 - 8 x 2 = 728) 5. The result is should be divisible by 7. (72 - 8 x 2 = 56) 56 is divisible by 7. Therefore, 745,514 is divisible by 14.
	15	Check if the number is divisible by 3 and 5.Check if the number is divisible by 3 and 5.	Take 2,955, for example, the sum of all digits is 21, and the last digit of the number is 5. Therefore, 2,955 is divisible by 15.Take 2,955, for example, the sum of all digits is 21, and the last digit of the number is 5. Therefore, 2,955 is divisible by 15.
	16	Similar to divisibility by 8, except check if the last 4 digits are divisible by 16 rather than the last 3.Alternatively, to make checking easier, check the thousands digit first. If it is an odd number, add 8 to the last three digits, and then check the last three digits. On the other hand, if it is an even number, simply check the last three digits directly.	2,295,632 is divisible by 16, because the last 4 digits is 5,632.Take 22,768, for example. Since the thousands digit is 2, check the last 3 digits directly. 768 is divisible by 16, so 22,768 is also divisible by 16. For another example, 37936, because the thousands digit is 9, 8 is added to the last 3 digits. 944 is divisible by 16.
	17	Take the last digit of the number, multiply it by five, and then subtract it to the remaining digits of the numbers. Repeat the process until the result can be easily identified.Adding 12 times the last digit to the remaining digits can also work, but it is a lot harder to calculate.	Take 59,789 for example. Take out 9 from 59789 and multiply it by five. (9 x 5 = 45) Subtract it from the remaining digits. (5978 - 45 = 5933) Continue repeating the same steps. (593 - 3 x 5 = 578; 57 - 8 x 5 = 17) 17 is divisible by 17, so 59,789 is also divisible by 17.Take 59,789 for example. Take out 9 from 59789 and multiply it by five. (9 x 5 = 45) Subtract it from the remaining digits. (5978 - 45 = 5933) Continue repeating the same steps. (593 - 3 x 5 = 578; 57 - 8 x 5 = 17) 17 is divisible by 17, so 59,789 is also divisible by 17.
	18	Because its prime factorization is 2 x 32, the number needs to be divisible by 2 and 9.Because its prime factorization is 2 x 32, the number needs to be divisible by 2 and 9.	671,634 is an even number, and the sum of all digits is 27. Therefore, 671,634 is divisible by 18.671,634 is an even number, and the sum of all digits is 27. Therefore, 671,634 is divisible by 18.
	19	Take the last digit of the number, multiply it by two, and then add it to the remaining digits of the numbers. Repeat the process until the result can be easily identified.Alternatively, take out 2 digits, and multiply the number by 4, and then adding it to the remaining digits. This is, however, not recommended, as the first method is a lot simpler.	Take 3,971 for example. Take out 1 from 3971 and multiply it by two. (1 x 2 = 2) Add it to the remaining digits. (397 + 2 = 399) Continue repeating the same steps. (39 + 9 x 2 = 57) 57 is divisible by 19, so 3,971 is also divisible by 19.Take 3,971 for example. Take out 1 from 3971 and multiply it by two. (1 x 2 = 2) Add it to the remaining digits. (397 + 2 = 399) Continue repeating the same steps. (39 + 9 x 2 = 57) 57 is divisible by 19, so 3,971 is also divisible by 19.
	20	Check the last 2 digits: They should end in 20, 40, 60, 80, or 00.Check the last 2 digits: They should end in 20, 40, 60, 80, or 00.	46,186,940 is divisible by 20 because the last 2 digits is divisible by 20.46,186,940 is divisible by 20 because the last 2 digits is divisible by 20.

Divisibility of Numbers between 21 and 30

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	21	Check the divisibility of 3 and 7.Alternatively, take the last digit of the number, multiply it by two, and then subtract it from the remaining digits of the numbers. Repeat the process until the result can be easily identified.	Take 73,857, for example: the sum of digits is 30, making it divisible by 3. Then, perform divisibility check of 7.Divisibility of 7: Take away the last digit of the number, multiply it by 2, and subtract it from the remaining digits. Repeat the process until the result can be determined. Example: For the number 73,857, 1. Take the last digit of the number and multiply it by 2. (7 x 2 = 14) 2. Subtract that number from the rest of the digits. (7385 - 14 = 7371) 3. Repeat the process. (737 - 1 x 2 = 735) 4. Continue until the results can be determined. (73 - 5 x 2 = 63) 5. The result is should be divisible by 7. (6 - 3 x 2 = 0) As 73,857 is divisible by both 3 and 7, 73,857 is divisible by 21.To be filled up
	22	Check the divisibility of 2 and 11.Check the divisibility of 2 and 11.	33,242 is an even number divisible by 11Divisibility of 11: Sum up the odd digits, then the even digits. Subtract them. Then check if the number is a multiple of 11. Example: For the number 33,242, the sum of all odd digits is 7, and the sum of all even digits is 7. After subtracting them, the result is 0, which is divisible by 11; therefore, 33,242 is also divisible by 11., so 33,242 is divisible by 22.33,242 is an even number divisible by 11Divisibility of 11: Sum up the odd digits, then the even digits. Subtract them. Then check if the number is a multiple of 11. Example: For the number 33,242, the sum of all odd digits is 7, and the sum of all even digits is 7. After subtracting them, the result is 0, which is divisible by 11; therefore, 33,242 is also divisible by 11., so 33,242 is divisible by 22.
	23	Take the last digit of the number, multiply it by seven, and then add it to the remaining digits of the numbers. Repeat the process until the result can be easily identified.Take the last digit of the number, multiply it by seven, and then add it to the remaining digits of the numbers. Repeat the process until the result can be easily identified.	Take 39,077 for example. Take out 7 from 39077 and multiply it by seven. (7 x 7 = 49) Add it to the remaining digits. (3907 + 49 = 3956) Repeat the same steps. (395 + 6x7 = 437; 43 + 7x7 = 92) 92 is divisible by 23, so 39,077 is also divisible by 23.Take 39,077 for example. Take out 7 from 39077 and multiply it by seven. (7 x 7 = 49) Add it to the remaining digits. (3907 + 49 = 3956) Repeat the same steps. (395 + 6x7 = 437; 43 + 7x7 = 92) 92 is divisible by 23, so 39,077 is also divisible by 23.
	24	Check the divisibility of 3 and 8.Check the divisibility of 3 and 8.	176,376 ends in 376 (divisible by 8), and the digit sum is 30 (divisible by 3). Therefore, 176,376 is divisible by 24.176,376 ends in 376 (divisible by 8), and the digit sum is 30 (divisible by 3). Therefore, 176,376 is divisible by 24.
	25	The last 2 digits ends in 25, 50, 75, or 00.The last 2 digits ends in 25, 50, 75, or 00.	149,346,775 is divisible by 25. The last two digits of 149346775 are 75, which is divisible by 25.149,346,775 is divisible by 25. The last two digits of 149346775 are 75, which is divisible by 25.
	26	Check the divisibility of 2 and 13.Check the divisibility of 2 and 13.	For the number 2,216,734: It is divisible by 2, since it is an even number. Perform divisibility check of 13.Divisibility of 13: Similar to divisibility of 7, but this time, take away the last digit of the number, multiply it by 4, and add it to the remaining digits. Then repeat the process. Example: For the number 2,216,734, 1. Take the last digit of the number and multiply it by 4. (4 x 4 = 16) 2. Add that number to the rest of the digits. (221673 + 16 = 221689) 3. Repeat the process. (22168 + 9 x 4 = 22204) 4. Continue until the results can be determined. (2220 + 4 x 4 = 2236) 5. The result should be divisible by 13. (223 + 6 x 4 = 247) 6. (24 + 7 x 4 = 52) Alternatively, the alternating sum method can be used instead. First, group the numbers of three from right to left. Then alternate between subtraction and addition. Then apply the above method to determine its divisibility of 13. Example: For number 107,359,993, 1. The alternating sum is 993 - 359 + 107 = 741 2. Examine the result. Use the 1st method again to reduce the digits if needed. 52 is divisible by 13. Therefore, 2,216,734 is divisible by 26.For the number 2,216,734: It is divisible by 2, since it is an even number. Perform divisibility check of 13.Divisibility of 13: Similar to divisibility of 7, but this time, take away the last digit of the number, multiply it by 4, and add it to the remaining digits. Then repeat the process. Example: For the number 2,216,734, 1. Take the last digit of the number and multiply it by 4. (4 x 4 = 16) 2. Add that number to the rest of the digits. (221673 + 16 = 221689) 3. Repeat the process. (22168 + 9 x 4 = 22204) 4. Continue until the results can be determined. (2220 + 4 x 4 = 2236) 5. The result should be divisible by 13. (223 + 6 x 4 = 247) 6. (24 + 7 x 4 = 52) Alternatively, the alternating sum method can be used instead. First, group the numbers of three from right to left. Then alternate between subtraction and addition. Then apply the above method to determine its divisibility of 13. Example: For number 107,359,993, 1. The alternating sum is 993 - 359 + 107 = 741 2. Examine the result. Use the 1st method again to reduce the digits if needed. 52 is divisible by 13. Therefore, 2,216,734 is divisible by 26.
	27	Add up the digits in blocks of three and check if it is divisible by 27. If the result is not easy to determine, subtract 8 times the last digit from the rest of the digits.Add up the digits in blocks of three and check if it is divisible by 27. If the result is not easy to determine, subtract 8 times the last digit from the rest of the digits.	Take 9,782,673,561, for example: Sum up the digits in blocks of three. (561 + 673 + 782 + 9 = 2025) Take out the last digit and multiply it by 8. (5 x 8 = 40) Subtract it to the rest of the digits (202 - 40 = 162) Repeat the last two steps. (16 - 2 x 8 = 0) Since 0 is divisible by 27, 9,782,673,561 is also divisible by 27.Take 9,782,673,561, for example: Sum up the digits in blocks of three. (561 + 673 + 782 + 9 = 2025) Take out the last digit and multiply it by 8. (5 x 8 = 40) Subtract it to the rest of the digits (202 - 40 = 162) Repeat the last two steps. (16 - 2 x 8 = 0) Since 0 is divisible by 27, 9,782,673,561 is also divisible by 27.
	28	Check the divisibility of 4 and 7.Check the divisibility of 4 and 7.	For the number 457,772: It is divisible by 4, since 72, the last two digits, is divisible by 4. Perform divisibility check of 7.Divisibility of 7: Take away the last digit of the number, multiply it by 2, and subtract it from the remaining digits. Repeat the process until the result can be determined. Example: For the number 457,772, 1. Take the last digit of the number and multiply it by 2. (2 x 2 = 4) 2. Subtract that number from the rest of the digits. (45777 - 4 = 45773) 3. Repeat the process. (4577 - 3 x 2 = 4571) 4. Continue until the results can be determined. (457 - 1 x 2 = 455) 5. The result is should be divisible by 7. (45 - 5 x 2 = 35) 35 is divisible by 7. Therefore, 457,772 is divisible by 28.For the number 457,772: It is divisible by 4, since 72, the last two digits, is divisible by 4. Perform divisibility check of 7.Divisibility of 7: Take away the last digit of the number, multiply it by 2, and subtract it from the remaining digits. Repeat the process until the result can be determined. Example: For the number 457,772, 1. Take the last digit of the number and multiply it by 2. (2 x 2 = 4) 2. Subtract that number from the rest of the digits. (45777 - 4 = 45773) 3. Repeat the process. (4577 - 3 x 2 = 4571) 4. Continue until the results can be determined. (457 - 1 x 2 = 455) 5. The result is should be divisible by 7. (45 - 5 x 2 = 35) 35 is divisible by 7. Therefore, 457,772 is divisible by 28.
	29	To check if a number is divisible by 29, take off the last digit, triple it, and then add it to the rest of the digits.To check if a number is divisible by 29, take off the last digit, triple it, and then add it to the rest of the digits.	Take 105,299, for example: Take out 9 from 105,299 and multiply it by three. (9 x 3 = 27) Add it from the remaining digits. (10529 + 27 = 10556) Repeat the same steps. (1055 + 6x3 = 1073; 107 + 3x3 = 116) 116 is divisible by 29, so 105,299 is also divisible by 29.Take 105,299, for example: Take out 9 from 105,299 and multiply it by three. (9 x 3 = 27) Add it from the remaining digits. (10529 + 27 = 10556) Repeat the same steps. (1055 + 6x3 = 1073; 107 + 3x3 = 116) 116 is divisible by 29, so 105,299 is also divisible by 29.
	30	Simply check the divisibility of 3 and 10.Simply check the divisibility of 3 and 10.	8146920 is divisible by 30 because the last digit is 0, and the sum of digits is 30.8146920 is divisible by 30 because the last digit is 0, and the sum of digits is 30.

Divisibility of Numbers between 31 and 40

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	31	To check if a number is divisible by 31, take off the last digit, triple it, and then subtract it to the rest of the digits.To check if a number is divisible by 31, take off the last digit, triple it, and then subtract it to the rest of the digits.	For the number 31,279, take out 9, triple it (27), and then subtract it from 3127 (3127-27=3100). 3,100 is divisible by 31, therefore 31,279 is also divisible by 31.For the number 31,279, take out 9, triple it (27), and then subtract it from 3127 (3127-27=3100). 3,100 is divisible by 31, therefore 31,279 is also divisible by 31.
	32	Check the last 5 digits, although if the ten thousands digit is an odd number, add 16 to the last 4 digits and check them; if the tens thousand digit is even, directly check the last 4 digits.Check the last 5 digits, although if the ten thousands digit is an odd number, add 16 to the last 4 digits and check them; if the tens thousand digit is even, directly check the last 4 digits.	For the number 1,913,248, the ten thousands digit is 1; therefore, add 16 to the last 4 digits, 3,248. Afterwards, the result 3264 is divisible by 32, so 1,913,248 is divisible by 32.For the number 1,913,248, the ten thousands digit is 1; therefore, add 16 to the last 4 digits, 3,248. Afterwards, the result 3264 is divisible by 32, so 1,913,248 is divisible by 32.
	33	Add up the digits in groups of two from right to left, and check its divisibility by 33. Or, check the divisibility of 3 and 11 separately.Alternatively, check its divisibility by 3 and 11 individually. Or, take out the last digit, multiply it by 10, and then add it to the rest to check divisibility by 33.	12,894,387 is divisible by 33 because the digit sum in groups of two is 87 + 43 + 89 + 12 = 231, divisible by 33.For the number 42,537, the digit sum is 21, divisible by 3. The alternating sum is 11, divisible by 11. Therefore, 42,537 is divisible by 33. Or, using the right trim method, (4253 + 70 = 4323; 432 + 30 = 462; 46 + 20 = 66), the result, 66, is divisible by 33.
	34	Check its divisibility to both 2 and 17.Check its divisibility to both 2 and 17.
	35	Check its divisibility to both 5 and 7.Check its divisibility to both 5 and 7.
	36	Check its divisibility to both 4 and 9.Check its divisibility to both 4 and 9.
	37	Add up the digits in blocks of three and check if it is divisible by 37. If the result is not easy to determine, subtract 11 times the last digit from the rest of the digits.Add up the digits in blocks of three and check if it is divisible by 37. If the result is not easy to determine, subtract 11 times the last digit from the rest of the digits.	Take 13,966,427, for example: Sum up the digits in blocks of three. (427 + 966 + 13 = 1406) Take out the last digit and multiply it by 11. (6 x 11 = 66) Subtract it to the rest of the digits (140 - 66 = 74) Since 74 is divisible by 37, 13,966,427 is also divisible by 37.Take 13,966,427, for example: Sum up the digits in blocks of three. (427 + 966 + 13 = 1406) Take out the last digit and multiply it by 11. (6 x 11 = 66) Subtract it to the rest of the digits (140 - 66 = 74) Since 74 is divisible by 37, 13,966,427 is also divisible by 37.
	38	Check its divisibility to both 2 and 19.Check its divisibility to both 2 and 19.
	39	Take out the last digit, multiply it by 4, and then add it to the rest of the digits.Take out the last digit, multiply it by 4, and then add it to the rest of the digits.	Take 140,673, for example: Take out 3 from 140,673 and multiply it by four. (3 x 4 = 12) Add it from the remaining digits. (14067 + 12 = 14079) Repeat the same steps. (1407 + 9x4 = 1443; 144 + 3x4 = 156) 156 is divisible by 39, so 140,673 is also divisible by 39.Take 140,673, for example: Take out 3 from 140,673 and multiply it by four. (3 x 4 = 12) Add it from the remaining digits. (14067 + 12 = 14079) Repeat the same steps. (1407 + 9x4 = 1443; 144 + 3x4 = 156) 156 is divisible by 39, so 140,673 is also divisible by 39.
	40	Check the last 3 digits and determine its divisibility by 40.Alternatively, check its divisibility to both 5 and 8.

Divisibility of Numbers between 41 and 50

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	41	Take out the last digit, multiply it by 4, and then subtract it from the remaining digits.Optionally, for extremely large numbers, add up all the digits in groups of five from right to left before applying the first method.	For the number 661,289, take out 9, quadruple it (36), and then subtract it from 66128 (66128-36=66092). Repeat the steps (6609-2x4=6601; 660-1x4=656; 65-6x4=41). 41 is divisible by 41, therefore 661,289 is also divisible by 41.For the number 661,289, take out 9, quadruple it (36), and then subtract it from 66128 (66128-36=66092). Repeat the steps (6609-2x4=6601; 660-1x4=656; 65-6x4=41). 41 is divisible by 41, therefore 661,289 is also divisible by 41.
	42	Check the divisibility of 2, 3, and 7.Check the divisibility of 2, 3, and 7.
	43	Take out the last digit, multiply it by 30, and then subtract it to the remaining digits.Alternatively, multiply the last digit by 13 and add it to the remaining digits instead. Or, taking out two digits, multiply it by 3, and then subtract it from the rest of the digits is also an option.	For the number 549,067: 54906 - 7x30 = 54696; 5469 - 6x30 = 5289; 528 - 9x30 = 258; 258 is divisible by 43, and therefore 549,067 is divisible by 43.For the number 549,067: 54906 - 7x30 = 54696; 5469 - 6x30 = 5289; 528 - 9x30 = 258; 258 is divisible by 43, and therefore 549,067 is divisible by 43.
	44	Check the divisibility of 4 and 11.Check the divisibility of 4 and 11.
	45	Check the divisibility of 5 and 9.Check the divisibility of 5 and 9.
	46	Check the divisibility of 2 and 23.Check the divisibility of 2 and 23.
	47	Take out the last digit, multiply it by 14, and then subtract it from the remaining digits.It is possible to add 33 times the last digit to the rest instead, but the difficulty of calculating it makes it less practical.
	48	Check the divisibility of 3 and 16.Check the divisibility of 3 and 16.
	49	Take out the last digit, multiply it by 5, and then add it to the remaining digits.Take out the last digit, multiply it by 5, and then add it to the remaining digits.
	50	Check the last 2 digits. It should end in 50 or 00.Check the last 2 digits. It should end in 50 or 00.	2800 is divisible by 50: the last two digits of 2800 are 0, which is divisible by 50.2800 is divisible by 50: the last two digits of 2800 are 0, which is divisible by 50.

Divisibility of Numbers between 51 and 60

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	51	Take out the last digit, multiply it by 5, and then subtract it from the remaining digits.Take out the last digit, multiply it by 5, and then subtract it from the remaining digits.
	52	Check the divisibility of 4 and 13.Check the divisibility of 4 and 13.
	53	Take out the last digit, multiply it by 16, and then add it to the remaining digits.It is also viable to take the last two digits, multiply it by 9, and then subtract it to the remaining digits.
	54	Check the divisibility of 2 and 27.Check the divisibility of 2 and 27.
	55	Check the divisibility of 5 and 11.Check the divisibility of 5 and 11.
	56	Check the divisibility of 7 and 8.Check the divisibility of 7 and 8.
	57	Check the divisibility of 3 and 19.Alternatively, take out the last two digits and multiply it by 4, then add it to the remaining digits. This is, however, much harder to calculate.
	58	Check the divisibility of 2 and 29.Check the divisibility of 2 and 29.
	59	Take out the last digit, multiply it by 6, and then add it to the remaining digits.Take out the last digit, multiply it by 6, and then add it to the remaining digits.
	60	Perform divisibility checks for 3 and 20.Perform divisibility checks for 3 and 20.

Divisibility of Numbers between 61 and 70

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	61	Take out the last digit, multiply it by 6, and then subtract it from the remaining digits.Take out the last digit, multiply it by 6, and then subtract it from the remaining digits.
	62	Check the divisibility of 2 and 31.Check the divisibility of 2 and 31.
	63	Check the divisibility of 7 and 9.Check the divisibility of 7 and 9.
	64	Check the last 6 digits.Check the last 6 digits.
	65	Perform divisibility checks for 5 and 13.Perform divisibility checks for 5 and 13.
	66	Perform divisibility checks for 6 and 11.Perform divisibility checks for 6 and 11.
	67	Take out the last digit, multiply it by 20, and then subtract it from the remaining digits.Alternatively, take out the last two digits, multiply it by 2, and then subtracting it from the remaining digits. Adding 47 times the last digit to the remaining digits is technically another method, but it is obviously a lot more difficult to perform.
	68	Perform divisibility checks for 4 and 17.Perform divisibility checks for 4 and 17.
	69	Take out the last digit, multiply it by 7, and then add it to the remaining digits. It is unnecessary to check divisibility of 3 and 23 individually.Take out the last digit, multiply it by 7, and then add it to the remaining digits. It is unnecessary to check divisibility of 3 and 23 individually.
	70	Perform divisibility checks for 7 and 10.Perform divisibility checks for 7 and 10.

Divisibility of Numbers between 71 and 80

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	71	Take out the last digit, multiply it by 7, and then subtract it from the remaining digits.Take out the last digit, multiply it by 7, and then subtract it from the remaining digits.
	72	Check the divisibility of 8 and 9.Check the divisibility of 8 and 9.
	73	Form the alternating sum of four digits as a group from right to left. Afterwards, take out the last digit, multiply it by 22, and then add it to the remaining digits.Form the alternating sum of four digits as a group from right to left. Afterwards, take out the last digit, multiply it by 22, and then add it to the remaining digits.
	74	Check the divisibility of 2 and 37.Check the divisibility of 2 and 37.
	75	Check the divisibility of 3 and 25.Check the divisibility of 3 and 25.
	76	Check the divisibility of 4 and 19.Check the divisibility of 4 and 19.
	77	Check the divisibility of 7 and 11.Check the divisibility of 7 and 11.
	78	Check the divisibility of 2 and 39.Check the divisibility of 2 and 39.
	79	Take out the last digit, multiply it by 8, and then add it to the remaining digits.Take out the last digit, multiply it by 8, and then add it to the remaining digits.
	80	Check the last 4 digits and determine its divisibility by 80.Check the last 4 digits and determine its divisibility by 80.

Divisibility of Numbers between 81 and 90

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	81	Take out the last digit, multiply it by 8, and then subtract it from the remaining digits.Take out the last digit, multiply it by 8, and then subtract it from the remaining digits.
	82	Check the divisibility of 2 and 41.Check the divisibility of 2 and 41.
	83	Take out the last digit, multiply it by 25, and then add it to the remaining digits.Take out the last digit, multiply it by 25, and then add it to the remaining digits.
	84	Check the divisibility of 3, 4, and 7.Check the divisibility of 3, 4, and 7.
	85	Check the divisibility of 5 and 17.Check the divisibility of 5 and 17.
	86	Check the divisibility of 2 and 43.Check the divisibility of 2 and 43.
	87	Check the divisibility of 3 and 29.Check the divisibility of 3 and 29.
	88	Check the divisibility of 8 and 11.Check the divisibility of 8 and 11.
	89	Take out the last digit, multiply it by 9, and then add it to the remaining digits.Take out the last digit, multiply it by 9, and then add it to the remaining digits.
	90	Check the divisibility of 9 and 10.Check the divisibility of 9 and 10.

Divisibility of Numbers between 91 and 100

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	91	Form the alternating sum in groups of three from right to left. Afterwards, take out the last digit, multiply it by 9, and then subtract it from the remaining digits.Form the alternating sum in groups of three from right to left. Afterwards, take out the last digit, multiply it by 9, and then subtract it from the remaining digits.
	92	Check the divisibility of 4 and 23.Check the divisibility of 4 and 23.
	93	Check the divisibility of 3 and 31.Check the divisibility of 3 and 31.
	94	Check the divisibility of 2 and 47.Check the divisibility of 2 and 47.
	95	Check the divisibility of 5 and 19.Check the divisibility of 5 and 19.
	96	Check the divisibility of 3 and 32 separately.Take out the leftmost digit and multiply it by 4. Move that number two spaces to the right, and then add it to the remaining digits.
	97	Take out the leftmost digit and multiply it by 3. Move that number two spaces to the right, and then add it to the remaining digits.Alternatively, using the right trim method, subtract 29 times the last digit to the rest of the digits. This is, however, a lot more difficult.
	98	Take out the leftmost digit and multiply it by 2. Move that number two spaces to the right, and then add it to the remaining digits.Check the divisibility of 2 and 49.
	99	Add up all the digits in groups of two from right to left.Alternatively, check the divisibility of 9 and 11 separately.
	100	Examine the last 2 digits. It is divisible by 100 when the number ends in 00.Examine the last 2 digits. It is divisible by 100 when the number ends in 00.

Divisibility of Numbers between 101 and 110

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	101	Take out the last digit, multiply it by 10, and then subtract it from the remaining digits.Alternatively, calculate the alternating sum in groups of two from right to left and determine if it is divisible by 101.
	102	Check the divisibility of 6 and 17.Check the divisibility of 6 and 17.
	103	Subtract the last two digits from 3 times the rest, and check if it is divisible by 103. If the result is not easy to determine, subtract 31 times the last digit from the rest of the digits.Subtract the last two digits from 3 times the rest, and check if it is divisible by 103. If the result is not easy to determine, subtract 31 times the last digit from the rest of the digits.
	104	Check the divisibility of 8 and 13.Check the divisibility of 8 and 13.
	105	Check the divisibility of 3, 5, and 7.Check the divisibility of 3, 5, and 7.
	106	Check the divisibility of 2 and 53.Check the divisibility of 2 and 53.
	107	Take out the last digit, multiply it by 32, and then subtract it from the remaining digits.Take out the last digit, multiply it by 32, and then subtract it from the remaining digits.
	108	Check the divisibility of 4 and 27.Check the divisibility of 4 and 27.
	109	Take out the last digit, multiply it by 11, and then add it to the remaining digits.Take out the last digit, multiply it by 11, and then add it to the remaining digits.
	110	Check the divisibility of 10 and 11.Check the divisibility of 10 and 11.

Divisibility of Numbers between 111 and 120

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	111	Add up all the digits in groups of three from right to left.Alternatively, check the divisibility of 3 and 37 separately.
	112	Check the divisibility of 7 and 16.Check the divisibility of 7 and 16.
	113	Take out the last digit, multiply it by 34, and then add it to the remaining digits.Take out the last digit, multiply it by 34, and then add it to the remaining digits.
	114	Check the divisibility of 6 and 19.Check the divisibility of 6 and 19.
	115	Check the divisibility of 5 and 23.Check the divisibility of 5 and 23.
	116	Check the divisibility of 4 and 29.Check the divisibility of 4 and 29.
	117	Check the divisibility of 9 and 13.Check the divisibility of 9 and 13.
	118	Check the divisibility of 2 and 59.Check the divisibility of 2 and 59.
	119	Check the divisibility of 7 and 17.Check the divisibility of 7 and 17.
	120	Check the divisibility of 3 and 40.Check the divisibility of 3 and 40.

Divisibility of Numbers between 121 and 130

	Divisor	Main Method (Easier)Alternative Method (Harder)	Examples (for Main Method)Examples (for Alternative Method)
		Primary methods Alternative methods
	121	Multiply the last digit by 12 and subtract it from the rest.Multiply the last digit by 12 and subtract it from the rest.
	122	Check if the number is divisible by 2 and 61.Check if the number is divisible by 2 and 61.
	123	Check if the number is divisible by 3 and 41.Check if the number is divisible by 3 and 41.
	124	Check if the number is divisible by 2 and 62.Check if the number is divisible by 2 and 62.
	125	The last three digits must be divisible by 125.The last three digits must be divisible by 125.
	126	Check if the number is divisible by 2 and 63.Check if the number is divisible by 2 and 63.
	127	Multiply the last digit by 38 and subtract it from the rest.Multiply the last digit by 38 and subtract it from the rest.
	128	The last seven digits must be divisible by 128.The last seven digits must be divisible by 128.
	129	Check if the number is divisible by 3 and 43.Check if the number is divisible by 3 and 43.
	130	Check if the number is divisible by 2 and 65.Check if the number is divisible by 2 and 65.

Techinical Divisibility of Number Information

Below lists down all the possible methods. The methods recommended will be bolded and displayed in blue text.

Prime numbers from 2 ~ 100

Divisor	Simple methods Advanced methods				Examples
	Last digit(s)Right trim (1 digit)	Digit sumRight trim (2 digits)	Blocks of digit sumRight trim (3 digits)	Combined methodsLeft trim
7	5,-2 Multiply the last digit by 2, and subtract it from the remaining digits. Or, alternatively, add 5 times the last digit to the remaining digits
11	-1, 10	(-10 , 1)	(-1 , 10)
13	-2, 5	(-10 , 3)	(-1 , 12)
17	-5, 12	(-9 , 8)	(-6 , 11)
19	-17, 2	(-15 , 4)	(-11 , 8)
23	-16, 7	(-20 , 3)	(-2 , 21)
29	-26, 3	(-20 , 9)	(-2 , 27)
31	-3, 28	(-22 , 9)	(-27 , 4)
37	-11, 26	(-27 , 10)	(-36 , 1)
41	-4, 37	(-25 , 16)	(-23 , 18)
43	-30, 13	(-3 , 40)	(-39 , 4)
47	-33, 14	(-39 , 8)	(-18 , 29)
53	-37, 16	(-9 , 44)	(-38 , 15)
59	-53, 6	(-23 , 36)	(-20 , 39)
61	-6, 55	(-25 , 36)	(-33 , 28)
67	-20, 47	(-2 , 65)	(-27 , 40)
71	-7, 64	(-22 , 49)	(-59 , 12)
73	-51, 22	(-27 , 46)	(-10 , 63)
79	-71, 8	(-15 , 64)	(-41 , 38)
83	-58, 25	(-39 , 44)	(-62 , 21)
89	-80, 9	(-8 , 81)	(-72 , 17)
97	-29, 68	(-32 , 65)	(-42 , 55)
101	-10, 91	(-1 , 100)	(-79 , 24)
103	-72, 31	(-69 , 34)	(-26 , 81)
107	-32, 75	(-46 , 61)	(-86 , 23)
109	-98, 11	(-97 , 12)	(-20 , 93)
113	-79, 34	(-87 , 26)

Important Pages
Divisibility and Sieves	Divisibility Rules (by rule) | (by list) | Sieve of Eratosthenes
Proofs and Theorems	Euclid's Proof | Goldbach's Conjecture | Prime Number Theorem | Integer
Prime Numbers	Timeline of the History of Prime Numbers | The Largest Known Prime Number
Help	Getting Started
Different Bases	What Digit a Prime Number in a Base Can End In