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## Divisibilities of 1 ~ 10

Divisibility of 1:
Every number is divisible by 1.
Example: 375 is divisible by 1.

Divisibility of 2:
All even numbers are divisible by 2, which are
numbers with 0, 2, 4, 6, or 8 as the last digit.
Example: 2,624 is divisible by 2. Its last digit is 4.

Divisibility of 3:
Sum the digits. If the result is divisible by 3,
then the number is divisible by 3.
Example: For the number 3,177, the sum
of all digits is 18. The result is divisible by 3;
therefore, 3,177 is also divisible by 3.

Divisibility of 4:
Check the last two digits.
Example: 3,756 is divisible by 4. The last two digits
of the number is 56, which is divisible by 4.

Divisibility of 5:
Check the last digit. Numbers with 0 or 5 as the last digit are divisible by 5.
Example: 35,515 is divisible by 5. Its last digit is 5.

Divisibility of 6:
Perform checks of divisibility of 2 and 3. If both
tests are positve, the number is divisible by 6.
Example: For the number 41,556, it is an
even number (last digit is 6), and the sum
of all digits is 21. The result is divisible by 3;
therefore, 41,556 is also divisible by 3.

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 172,781,
1. Take the last digit of the number and multiply it by 2. (1 x 2 = 2)
2. Subtract that number from the rest of the digits. (17278 - 2 = 17276)
3. Repeat the process. (1727 - 6 x 2 = 1715)
4. Continue until the results can be determined. (171 - 5 x 2 = 161)
5. The result is should be divisible by 7. (16 - 1 x 2 = 14)

Divisibility of 8:
Check the last three digits. Alternatively, if the third
digit is even, directly examine the last 2 digits. If it is
odd, add 4 to the last 2 digits and examine.
Example: 2,168 is divisible by 4. The last three
digits of the number is 168, which is divisible by 8.

Divisibility of 9:
Sum the digits. If the result is divisible by 9,
then the number is divisible by 9.
Example: For the number 26,613, the sum
of all digits is 18. The result is divisible by 9;
therefore, 26,613 is also divisible by 9.

Divisibility of 10:
The last digit is 0.
Example: 7,725,610 is divisible by 10. Its last digit is 0.

## Divisibilities of 11 ~ 20

Divisibility 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 38,621, the sum of all odd
digits is 10, and the sum of all even digits is 10. After
subtracting them, the result is 0, which is divisible by
11; therefore, 38,621 is also divisible by 11.

Divisibility of 12:
Perform checks of divisibility of 4 and 3. If both
tests are positve, the number is divisible by 12.
Example: For the number 857,676, the sum of
all digits is 39, divisible by 3; the last two digits
of the number is 76, which is divisible by 4.

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 1,864,369,
1. Take the last digit of the number and multiply it by 4. (9 x 4 = 36)
2. Add that number to the rest of the digits. (186436 + 36 = 186472)
3. Repeat the process. (18647 + 2 x 4 = 18655)

4. Continue until the results can be determined. (1865 + 5 x 4 = 1885)
5. The result should be divisible by 13. (188 + 5 x 4 = 208)
6. (20 + 8 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.

Divisibility of 14:
Perform checks of divisibility of 2 and 7. If both
tests are positve, the number is divisible by 14.

Divisibility of 15:
Perform checks of divisibility of 3 and 5. If both
tests are positve, the number is divisible by 15.

Divisibility of 16:
If the thousands digit is even, examine if the last
three digits of the number is divisible by 16.
If the thousands digit is odd, add 8 to the number
first, and then examine the last three digits of it.

Divisibility of 17:
Take away the last digit of the number, multiply it by 5, and subtract it from
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 238,493,
1. Take the last digit of the number and multiply it by 5. (3 x 5 = 15)
2. Subtract that number from the rest of the digits. (23849 - 15 = 23834)
3. Repeat the process. (2383 - 4 x 5 = 2363)

4. Continue until the results can be determined. (236 - 3 x 5 = 221)
5. The result is should be divisible by 17. (22 - 1 x 5 = 17)

Divisibility of 18:
Perform checks of divisibility of 2 and 9. If both
tests are positve, the number is divisible by 18.

Divisibility of 19:
Take away the last digit of the number, multiply it by 2, and add it to
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 977,759,
1. Take the last digit of the number and multiply it by 2. (9 x 2 = 18)
2. Add that number to the rest of the digits. (97775 + 18 = 97793)
3. Repeat the process. (9779 + 3 x 2 = 9785)

4. Continue until the results can be determined. (978 + 5 x 2 = 988)
5. The result is should be divisible by 19. (98 + 8 x 2 = 114)

Divisibility of 20:
The number formed by the last two digits is divisible by 20.

## Divisibilities of 21 ~ 30

Divisibility of 21:
Perform checks of divisibility of 3 and 7. If both
tests are positve, the number is divisible by 21.

Divisibility of 22:
Perform checks of divisibility of 2 and 11. If both
tests are positve, the number is divisible by 22.

Divisibility of 23:
Take away the last digit of the number, multiply it by 7, and add it to
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 953,741,
1. Take the last digit of the number and multiply it by 7. (1 x 7 = 7)
2. Add that number to the rest of the digits. (95374 + 7 = 95381)
3. Repeat the process. (9538 + 1 x 7 = 9545)

4. Continue until the results can be determined. (954 + 5 x 7 = 989)
5. The result is should be divisible by 23. (98 + 9 x 7 = 161)

Divisibility of 24:
It is most efficient to check the divisibility of
24 by checking its divisibility by 3 and 8.

Divisibility of 25:
Check the last two digits. They should end in
00, 25, 50, 75, which are divisible by 25.
Example: 1,435,675 is divisible by 25 due to
the last two digits of the number being 75.

Divisibility of 26:
Perform checks of divisibility of 2 and 13. If both
tests are positive, the number is divisible by 26.

Divisibility of 27:
Sum the digits in blocks of three and see if the result is divisible by 27.
Example: For number 8,435,529,
1. The digit sum in groups of 3 is 529 + 435 + 8 = 972
2. Examine the result.
If it cannot be determined, use the right trim method:
3. Multiply the last digit by 8 and subtract it from the rest. (97 - 2 x 8 = 81)

Divisibility of 28:
Perform checks of divisibility of 4 and 7. If both
tests are positive, the number is divisible by 28.

Divisibility of 29:
Take away the last digit of the number, multiply it by 3, and add it to
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 120,437,
1. Take the last digit of the number and multiply it by 2. (7 x 3 = 21)
2. Add that number to the rest of the digits. (12043 + 21 = 12064)
3. Repeat the process. (1206 + 4 x 3 = 1218)

4. Continue until the results can be determined. (121 + 8 x 3 = 145)
5. The result is should be divisible by 29. (14 + 5 x 3 = 29)

Divisibility of 30:
Perform checks of divisibility of 3 and 10. If both
tests are positive, the number is divisible by 30.

## Divisibilities of 31 ~ 50

Divisibility of 31:
Take away the last digit of the number, multiply it by 3, and subtract it from
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 707,699,
1. Take the last digit of the number and multiply it by 3. (9 x 3 = 27)
2. Subtract that number from the rest of the digits. (70769 - 27 = 70742)
3. Repeat the process. (7074 - 2 x 3 = 7068)

4. Continue until the results can be determined. (706 - 8 x 3 = 682)
5. The result is should be divisible by 31. (68 - 2 x 3 = 62)

Divisibility of 32:
The number formed by the last five digits is divisible by 32.
If the ten thousands digit is even, examine the number
formed by the last four digits.
If the ten thousands digit is odd, examine the number
formed by the last four digits plus 16.

Divisibility of 33:
Perform checks of divisibility of 3 and 11. If both
tests are positve, the number is divisible by 33.

Divisibility of 34:
Perform checks of divisibility of 2 and 17. If both
tests are positve, the number is divisible by 34.

Divisibility of 35:
Perform checks of divisibility of 5 and 7. If both
tests are positve, the number is divisible by 35.

Divisibility of 36:
Perform checks of divisibility of 4 and 9. If both
tests are positve, the number is divisible by 36.

Divisibility of 37:
Sum the digits in blocks of three and see if the result is divisible by 37.
Example: For number 6,132,593,009,
1. The digit sum in groups of 3 is 9 + 593 + 132 + 6 = 740
2. Examine the result.
If it cannot be determined, use the right trim method:
3. Multiply the last digit by 11 and subtract it from the rest. (74 - 0 x 11 = 74)

Divisibility of 38:
Perform checks of divisibility of 2 and 19. If both
tests are positve, the number is divisible by 38.

Divisibility of 39:
Perform checks of divisibility of 3 and 13. If both
tests are positve, the number is divisible by 39.

Divisibility of 40:
The number formed by the last three digits is divisible by 40.

Divisibility of 41:
Take away the last digit of the number, multiply it by 4, and subtract it from
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 948,863,
1. Take the last digit of the number and multiply it by 4. (3 x 4 = 12)
2. Subtract that number from the rest of the digits. (94886 - 12 = 94874)
3. Repeat the process. (9487 - 4 x 4 = 9471)

4. Continue until the results can be determined. (947 - 1 x 4 = 943)
5. The result is should be divisible by 41. (94 - 3 x 4 = 82)

Divisibility of 42:
Perform checks of divisibility of 2, 3, and 7. If all
tests are positive, the number is divisible by 42.

Divisibility of 43:
Take away the last digit of the number, multiply it by 30, and subtract it from
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 18,533,
1. Take the last digit of the number and multiply it by 30. (3 x 30 = 90)
2. Subtract that number from the rest of the digits. (1853 - 90 = 1763)
3. Repeat the process. (176 - 3 x 30 = 86)

Alternatively, one can also multiply the last 2 digits by 3,
and then subtracting it from the number. Then repeat the process.

Divisibility of 44:
Perform checks of divisibility of 4 and 11. If both
tests are positive, the number is divisible by 44.

Divisibility of 45:
The number must be divisible by 9 ending in 0 or 5.

Divisibility of 46:
Perform checks of divisibility of 2 and 23. If both
tests are positive, the number is divisible by 46.

Divisibility of 47:
Take away the last digit of the number, multiply it by 14, and subtract it from
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 802,901,
1. Take the last digit of the number and multiply it by 14. (1 x 14 = 14)
2. Subtract that number from the rest of the digits. (80290 - 14 = 80276)
3. Repeat the process. (8027 - 6 x 14 = 7943)

4. Continue until the results can be determined. (794 - 3 x 14 = 752)
5. The result is should be divisible by 47. (75 - 2 x 14 = 47)

Divisibility of 48:
Perform checks of divisibility of 3 and 16. If both
tests are positive, the number is divisible by 48.

Divisibility of 49:
Take away the last digit of the number, multiply it by 5, and add it to
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 313,257,
1. Take the last digit of the number and multiply it by 5. (7 x 5 = 35)
2. Subtract that number from the rest of the digits. (31325 + 35 = 31360)
3. Repeat the process. (3136 + 0 x 5 = 3136)

4. Continue until the results can be determined. (313 + 6 x 5 = 343)
5. The result is should be divisible by 47. (34 + 3 x 5 = 49)

Divisibility of 50:
The last two digits are 00 or 50.
Example: 2,437,850 is divisible by 50. Its last two digits are 50.

## Proofs of 1 ~ 10 Divisibility

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.

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.

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.

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.

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.

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.

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.

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