Divisibility rules

These divisibility rules help determine when positive integers are divisible by particular other integers. All of these rules apply for base-10 only -- other bases have their own, different versions of these rules.

1 Divisibility Rule for 2 and Powers of 2
2 Divisibility Rule for 3 and 9
3 Divisibility Rule for 5 and Powers of 5
4 Divisibility Rule for 7
5 Divisibility Rule for 11
6 Divisibility Rule for 13
7 More general note for primes
8 More general note for composites
- 8.1 Example
- 8.2 Solution
9 Example Problems
10 Resources
- 10.1 Books
- 10.2 Classes
11 See also

Divisibility Rule for 2 and Powers of 2

A number is divisible by $2^n$ if and only if the last ${n}$ digits of the number are divisible by $2^n$ .

Proof

Divisibility Rule for 3 and 9

A number is divisible by 3 or 9 if and only if the sum of its digits is divisible by 3 or 9, respectively. Note that this does not work for higher powers of 3. For instance, the sum of the digits of 1899 is divisible by 27, but 1899 is not itself divisible by 27.

Proof

Divisibility Rule for 5 and Powers of 5

A number is divisible by $5^n$ if and only if the last $n$ digits are divisible by that power of 5.

Proof

Divisibility Rule for 7

Rule 1: Partition $N$ into 3 digit numbers from the right ( $d_3d_2d_1,d_6d_5d_4,\dots$ ). The alternating sum ( $d_3d_2d_1 - d_6d_5d_4 + d_9d_8d_7 - \dots$ ) is divisible by 7 if and only if $N$ is divisible by 7.

Proof

Rule 2: Truncate the last digit of $N$ , double that digit, and subtract it from the rest of the number (or vice-versa). $N$ is divisible by 7 if and only if the result is divisible by 7.

Proof

Divisibility Rule for 11

A number is divisible by 11 if and only if the alternating sum of the digits is divisible by 11.

Proof

Divisibility Rule for 13

Rule 1: Truncate the last digit, multiply it by 4 and add it to the rest of the number. The result is divisible by 13 if and only if the original number was divisble by 13. This process can be repeated for large numbers, as with the second divisibility rule for 7.

Proof

Rule 2: Partition $N$ into 3 digit numbers from the right ( $d_3d_2d_1,d_6d_5d_4,\dots$ ). The alternating sum ( $d_3d_2d_1 - d_6d_5d_4 + d_9d_8d_7 - \dots$ ) is divisible by 13 if and only if $N$ is divisible by 13.

Proof

More general note for primes

For every prime number other than 2 and 5, there exists a rule similar to rule 2 for divisibility by 7. For a general prime $p$ , there exists some number $q$ such that an integer is divisible by $p$ if and only if truncating the last digit, multiplying it by $q$ and subtracting it from the remaining number gives us a result divisible by $p$ . Divisibility rule 2 for 7 says that for $p = 7$ , $q = 2$ . The divisibility rule for 11 is equivalent to choosing $q = 1$ . The divisibility rule for 3 is equivalent to choosing $q = -1$ . These rules can also be found under the appropriate conditions in number bases other than 10. Also note that these rules exist in two forms: if $q$ is replaced by $p - q$ then subtraction may be replaced with addition. We see one instance of this in the divisibility rule for 13: we could multiply by 9 and subtract rather than multiplying by 4 and adding.

More general note for composites

A number is divisible by $N$ , where the prime factorization of $N$ is $p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n}$ , if the number is divisible by each of $p_1^{e_1}, p_2^{e_2},\ldots, p_n^{e_n}$ .