Part 6: Number Theory

## 6.1: Divisibility tests

Notes

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.

Practice Exercises

Do the following exercises to practice applying divisibility tests.