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.
Contents
- 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 Divisibility Rule for 17
- 8 Divisibility Rule for 19
- 9 More general note for primes
- 10 More general note for composites
- 11 Example Problems
- 12 Resources
- 13 See also
Divisibility Rule for 2 and Powers of 2
A number is divisible by if and only if the last
digits of the number are divisible by
. Thus, in particular, a number is divisible by 2 if and only if its units digit is divisble by 2, i.e. if the number ends in 0, 2, 4, 6 or 8.
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.
Divisibility Rule for 5 and Powers of 5
A number is divisible by if and only if the last
digits are divisible by that power of 5.
Divisibility Rule for 7
Rule 1: Partition into 3 digit numbers from the right (
). The alternating sum (
) is divisible by 7 if and only if
is divisible by 7.
Rule 2: Truncate the last digit of , double that digit, and subtract it from the rest of the number (or vice-versa).
is divisible by 7 if and only if the result is divisible by 7.
Rule 3: "Tail-End divisibility." Note. This only tells you if it is divisible and NOT the remainder. Take a number say 12345. Look at the last digit and add or subtract a multiple of 7. In this case we get 12370 or 12310. Lop off the ending 0's and repeat. 1237 ==> 123 ==> 130 ==> 13 NOPE. Works in general with numbers that are relatively prime to the base (and works GREAT in binary).
Divisibility Rule for 11
A number is divisible by 11 if and only if the alternating sum of the digits is divisible by 11.
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.
Rule 2: Partition into 3 digit numbers from the right (
). The alternating sum (
) is divisible by 13 if and only if
is divisible by 13.
Divisibility Rule for 17
Truncate the last digit, multiply it by 5 and subtract from the remaining leading number. The number is divisible if and only if the result is divisible. The process can be repeated for any number.
Divisibility Rule for 19
Truncate the last digit, multiply it by 2 and add to the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers.
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 , there exists some number
such that an integer is divisible by
if and only if truncating the last digit, multiplying it by
and subtracting it from the remaining number gives us a result divisible by
. Divisibility rule 2 for 7 says that for
,
. The divisibility rule for 11 is equivalent to choosing
. The divisibility rule for 3 is equivalent to choosing
. 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
is replaced by
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 , where the prime factorization of
is
, if the number is divisible by each of
.
Example
Is 55682168544 divisible by 36?
Solution
First, we find the prime factorization of 36 to be . Thus we must check for divisibility by 4 and 9 to see if it's divisible by 36.
Since the last two digits, 44, of the number is divisible by 4, so is the entire number.
To check for divisibility by 9, we look to see if the sum of the digits is divisible by 9. The sum of the digits is 54 which is divisible by 9.
Thus, the number is divisible by both 4 and 9 and must be divisible by 36.
Example Problems
Resources
Books
- The AoPS Introduction to Number Theory by Mathew Crawford.
- The Art of Problem Solving by Sandor Lehoczky and Richard Rusczyk.
Classes