Divisibility Rules: A Review

Recall from your grade school mathematics that the division algorithm for whole numbers has 4 basic components:

• the dividend - which is the number to be divided,
• the divisor - which is the number that will the divide the dividend,
• the quotient - which is the answer to a division problem, and
• the remainder - which is what remains after the division is executed

The concept of "divisibility" arises from the fact that every number can be divided evenly by a unique set of numbers. This means such a number can be divided with a remainder of zero. Consider, for instance, the number 1,456. Suppose it is divided by the numbers 2, 3, 4, 5, and 6. The quotients and remainders are as follows:

• 1,456 / 2 = 728 r 0
• 1,456 / 3 = 485 r 1
• 1,456 / 4 = 364 r 0
• 1,456 / 5 = 291 r 1
• 1,456 / 6 = 242 r 4

From the results, we can say that 1,456 is divisible by 2 and 4 because of the zero remainder, but not by 3, 5, and 6 because of the non-zero remainders.

The divisibility rules that follow enable us to see if a number is divisible by another number without actually dividing. Aside from these rules, having stock knowledge of your multiplication tables will come in handy.


Divisibility by 1 - All whole numbers are divisible by 1. Recall that any number divided by 1 results in that same number.

Divisibility by 2 - All even numbers are divisible by 2.

Divisibility by 3 - A number is divisible by 3 if the sum of its digits is a multiple of 3.

Divisibility by 4 - Look at the last 2 digits. If they are a multiple of 4 or are zeros, then the entire number is divisible by 4.

Divisibility by 5 - A number is divisible by 5 if the last digit is either 0 or 5.

Divisibility by 6 - An even number divisible by 3 is divisible by 6.

Divisibility by 7 -

• Take the last digit in a number.
• Double and subtract the last digit in your number from the rest of the digits.
• Repeat the process for larger numbers.
• Example: 357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7.

Divisibility by 8 - Look at the last 3 digits. If they are a multiple of 8 or are zeros, then the entire number is divisible by 8.

Divisibility by 9 - A number is divisible by 9 if the sum of its digits is a multiple of 9.

Divisibility by 10 - All numbers that end in 0 are divisible by 10.

Divisibility by 11 -

• Starting with the left, take every second digit and find their sum.
• Take the sum of the remaining digits.
• Subtract the sum in step 2 from the sum in step 1.
• If the result in step 3 is zero or is divisible by 11, then the entire number is divisible by 11.

Divisibility by 12 - A number divisible by both 3 and 4 is divisible by 12.


Exercises:

• For each item, determine if the first number is divisible by the second number. 
• Then confirm your answer by doing the division.

1)   5,000; 8
2)  18,931; 11
3)   858; 4
4)  8,099; 7
5)  111,111,111; 9
6)  7,428; 12
7)  5,571; 3
8)  1,001; 7
9)  3,008; 8
10) 8,349; 11