6.1: Divisibility tests

A counting number, A, is said to be divisible by another counting number, B, if A divides B evenly with no remainder (i.e., A ÷ B = C, where C is another counting number). Here are the most common divisibility tests for counting numbers. Divisibility tests for 7 are a little convoluted, so we’ll skip those.

Divisibility test for 2

• Test: If the ones digit is 0, 2, 4, 6, or 8, then the number is divisible by 2.
• Example 1: 1440 is divisible by 2 because the ones digit is 0. [Check: 1440 ÷ 2 = 720.]
• Example 2: 1441 is not divisible by 2 because the ones digit is 1.

Divisibility test for 3

• Test: If the sum of the digits is divisible by 3, then the number is divisible by 3.
• Example 1: 1440 is divisible by 3 because 1 + 4 + 4 + 0 = 9, which is divisible by 3. [Check: 1440 ÷ 3 = 480.]
• Example 2: 1441 is not divisible by 3 because 1 + 4 + 4 + 1 = 10, which is not divisible by 3.

Divisibility test for 4

• Test: If the number formed by the tens and ones digits is divisible by 4, then the number is divisible by 4.
• Example 1: 1440 is divisible by 4 because 40 is divisible by 4. [Check: 1440 ÷ 4 = 360.]
• Example 2: 1441 is not divisible by 4 because 41 is not divisible by 4.

Divisibility test for 5

• Test: If the ones digit is 0 or 5, then the number is divisible by 5.
• Example 1: 1440 is divisible by 5 because the ones digit is 0. [Check: 1440 ÷ 5 = 288.]
• Example 2: 1441 is not divisible by 5 because the ones digit is 1.

Divisibility test for 6

• Test: If the ones digit is 0, 2, 4, 6, or 8, and the sum of the digits is divisible by 3, then the number is divisible by 6. [This combines the divisibility tests for 2 and 3.]
• Example 1: 1440 is divisible by 6 because it is divisible by both 2 and 3 (see above). [Check: 1440 ÷ 6 = 240.]
• Example 2: 1441 is not divisible by 6 because it is not divisible by both 2 and 3.

Divisibility test for 8

• Test: If the number formed by the hundreds, tens, and ones digits is divisible by 8, then the number is divisible by 8.
• Example 1: 1440 is divisible by 8 because 440 is divisible by 8. [Check: 1440 ÷ 8 = 180.]
• Example 2: 1441 is not divisible by 8 because 441 is not divisible by 8.

Divisibility test for 9

• Test: If the sum of the digits is divisible by 9, then the number is divisible by 9.
• Example 1: 1440 is divisible by 9 because 1 + 4 + 4 + 0 = 9, which is divisible by 9. [Check: 1440 ÷ 9 = 160.]
• Example 2: 1441 is not divisible by 9 because 1 + 4 + 4 + 1 = 10, which is not divisible by 9.

Divisibility test for 10

• Test: If the ones digit is 0, then the number is divisible by 10.
• Example 1: 1440 is divisible by 10 because the ones digit is 0. [Check: 1440 ÷ 10 = 144.]
• Example 2: 1441 is not divisible by 10 because the ones digit is 1.

The video below works through some examples of applying divisibility tests.