Rules of Divisibility

Mathematics

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A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardnerexplained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American

For divisors with multiple rules, the rules are generally ordered first for those appropriate for numbers with many digits, then those useful for numbers with fewer digits.
Note: To test divisibility by any number that can be expressed as 2n or 5n, in which n is a positive integer, just examine the last n digits.
Note: To test divisibility by any number expressed as the product of prime factors {\displaystyle p_{1}^{n}p_{2}^{m}p_{3}^{q}}
, we can separately test for divisibility by each prime to its appropriate power. For example, testing divisibility by 24 (24 = 8*3 = 23*3) is equivalent to testing divisibility by 8 (23) and 3 simultaneously, thus we need only show divisibility by 8 and by 3 to prove divisibility by 24.

The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious; for others (such as examining the last n digits) the result must be examined by other means.

For divisors with multiple rules, the rules are generally ordered first for those appropriate for numbers with many digits, then those useful for numbers with fewer digits.
Note: To test divisibility by any number that can be expressed as 2n or 5n, in which n is a positive integer, just examine the last n digits.
Note: To test divisibility by any number expressed as the product of prime factors {\displaystyle p_{1}^{n}p_{2}^{m}p_{3}^{q}}
, we can separately test for divisibility by each prime to its appropriate power. For example, testing divisibility by 24 (24 = 8*3 = 23*3) is equivalent to testing divisibility by 8 (23) and 3 simultaneously, thus we need only show divisibility by 8 and by 3 to prove divisibility by 24.

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First, take any number (for this example it will be 376) and note the last digit in the number, discarding the other digits. Then take that digit (6) while ignoring the rest of the number and determine if it is divisible by 2. If it is divisible by 2, then the original number is divisible by 2.

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Example
376 (The original number)
37 6 (Take the last digit)
6 ÷ 2 = 3 (Check to see if the last digit is divisible by 2)
376 ÷ 2 = 188 (If the last digit is divisible by 2, then the whole number is divisible by 2)

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Ex.
492 (The original number)
4 + 9 + 2 = 15 (Add each individual digit together)
15 is divisible by 3 at which point we can stop. Alternatively we can continue using the same method if the number is still too large:
1 + 5 = 6 (Add each individual digit together)
6 ÷ 3 = 2 (Check to see if the number received is divisible by 3)
492 ÷ 3 = 164 (If the number obtained by using the rule is divisible by 3, then the whole number is divisible by 3)

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First, take any number (for this example it will be 492) and add together each digit in the number (4 + 9 + 2 = 15). Then take that sum (15) and determine if it is divisible by 3. The original number is divisible by 3 (or 9) if and only if the sum of its digits is divisible by 3 (or 9).
If a number is a multiplication of 3 consecutive numbers then that number is always divisible by 3. This is useful for when the number takes the form of (n × (n − 1) × (n + 1))

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Ex.
336 (The original number)
6 × 7 × 8 = 336
336 ÷ 3 = 112

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The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4;[2][3] this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4. If any number ends in a two digit number that you know is divisible by 4 (e.g. 24, 04, 08, etc.), then the whole number will be divisible by 4 regardless of what is before the last two digits.

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Ex.
492 (The original number)
4 + 9 + 2 = 15 (Add each individual digit together)
15 is divisible by 3 at which point we can stop. Alternatively we can continue using the same method if the number is still too large:
1 + 5 = 6 (Add each individual digit together)
6 ÷ 3 = 2 (Check to see if the number received is divisible by 3)
492 ÷ 3 = 164 (If the number obtained by using the rule is divisible by 3, then the whole number is divisible by 3)

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Ex.
336 (The original number)
6 × 7 × 8 = 336
336 ÷ 3 = 112

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The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4;[2][3] this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4. If any number ends in a two digit number that you know is divisible by 4 (e.g. 24, 04, 08, etc.), then the whole number will be divisible by 4 regardless of what is before the last two digits.

Alternatively, one can simply divide the number by 2, and then check the result to find if it is divisible by 2. If it is, the original number is divisible by 4. In addition, the result of this test is the same as the original number divided by 4.

General rule
2092 (The original number)
20 92 (Take the last two digits of the number, discarding any other digits)
92 ÷ 4 = 23 (Check to see if the number is divisible by 4)
2092 ÷ 4 = 523 (If the number that is obtained is divisible by 4, then the original number is divisible by 4)

Divisibility by 5 is easily determined by checking the last digit in the number (475), and seeing if it is either 0 or 5. If the last number is either 0 or 5, the entire number is divisible by 5.[2][3]
If the last digit in the number is 0, then the result will be the remaining digits multiplied by 2. For example, the number 40 ends in a zero (0), so take the remaining digits (4) and multiply that by two (4 × 2 = 8). The result is the same as the result of 40 divided by 5(40/5 = 8).
If the last digit in the number is 5, then the result will be the remaining digits multiplied by two (2), plus one (1). For example, the number 125 ends in a 5, so take the remaining digits (12), multiply them by two (12 × 2 = 24), then add one (24 + 1 = 25). The result is the same as the result of 125 divided by 5 (125/5=25).

Alternative example
1720 (The original number)
1720 ÷ 2 = 860 (Divide the original number by 2)
860 ÷ 2 = 430 (Check to see if the result is divisible by 2)
1720 ÷ 4 = 430 (If the result is divisible by 2, then the original number is divisible by 4)

110 (The original number)
11 0 (Take the last digit of the number, and check if it is 0 or 5)
11 0 (If it is 0, take the remaining digits, discarding the last)
11 × 2 = 22 (Multiply the result by 2)
110 ÷ 5 = 22 (The result is the same as the original number divided by 5)

Divisibility by 5 is easily determined by checking the last digit in the number (475), and seeing if it is either 0 or 5. If the last number is either 0 or 5, the entire number is divisible by 5.[2][3]
If the last digit in the number is 0, then the result will be the remaining digits multiplied by 2. For example, the number 40 ends in a zero (0), so take the remaining digits (4) and multiply that by two (4 × 2 = 8). The result is the same as the result of 40 divided by 5(40/5 = 8).
If the last digit in the number is 5, then the result will be the remaining digits multiplied by two (2), plus one (1). For example, the number 125 ends in a 5, so take the remaining digits (12), multiply them by two (12 × 2 = 24), then add one (24 + 1 = 25). The result is the same as the result of 125 divided by 5 (125/5=25).

110 (The original number)
11 0 (Take the last digit of the number, and check if it is 0 or 5)
11 0 (If it is 0, take the remaining digits, discarding the last)
11 × 2 = 22 (Multiply the result by 2)
110 ÷ 5 = 22 (The result is the same as the original number divided by 5)

85 (The original number)
8 5 (Take the last digit of the number, and check if it is 0 or 5)
8 5 (If it is 5, take the remaining digits, discarding the last)
8 × 2 = 16 (Multiply the result by 2)
16 + 1 = 17 (Add 1 to the result)
85 ÷ 5 = 17 (The result is the same as the original number divided by 5)

Divisibility by 6 is determined by checking the original number to see if it is both an even number (divisible by 2) and divisible by 3.[6] This is the best test to use.
If the number is divisible by six, take the original number (246) and divide it by two (246 ÷ 2 = 123). Then, take that result and divide it by three (123 ÷ 3 = 41). This result is the same as the original number divided by six (246 ÷ 6 = 41).

Finding a remainder of a number when divided by 6
(1, −2, −2, −2, −2, and −2 goes on for the rest) No period. -- Minimum magnitude sequence
(1, 4, 4, 4, 4, and 4 goes on for the rest) -- Positive sequence
Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on.
Next, compute the sum of all the values and take the remainder on division by 6.
Example: What is the remainder when 1036125837 is divided by 6?
Multiplication of the rightmost digit = 1 × 7 = 7
Multiplication of the second rightmost digit = 3 × −2 = −6
Third rightmost digit = −16
Fourth rightmost digit = −10
Fifth rightmost digit = −4
Sixth rightmost digit = −2
Seventh rightmost digit = −12
Eighth rightmost digit = −6
Ninth rightmost digit = 0
Tenth rightmost digit = −2
Sum = −51
−51 ≡ 3 (mod 6)
Remainder = 3

Finding a remainder of a number when divided by 6
(1, −2, −2, −2, −2, and −2 goes on for the rest) No period. -- Minimum magnitude sequence
(1, 4, 4, 4, 4, and 4 goes on for the rest) -- Positive sequence
Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on.
Next, compute the sum of all the values and take the remainder on division by 6.
Example: What is the remainder when 1036125837 is divided by 6?
Multiplication of the rightmost digit = 1 × 7 = 7
Multiplication of the second rightmost digit = 3 × −2 = −6
Third rightmost digit = −16
Fourth rightmost digit = −10
Fifth rightmost digit = −4
Sixth rightmost digit = −2
Seventh rightmost digit = −12
Eighth rightmost digit = −6
Ninth rightmost digit = 0
Tenth rightmost digit = −2
Sum = −51
−51 ≡ 3 (mod 6)
Remainder = 3

By Mertcan YESILDAL
Student

Kadri Saman MTSO Vocational Technical Anatolian High School.