# Divisible

When a dividend is divided by a divisor, and the quotient is a whole number with no remainder, the dividend is said to be divisible by the divisor.

The figure below shows that 8 is divisible by 2, but not 3. On the left, we can see that 8 can be evenly divided into 4 groups of 2. On the other hand, 8 cannot be evenly divided into 3 groups. Only 2 groups can contain 3 objects while the third group can only contain 2 objects.

Examples

Determine whether the following are divisible.

1. 48 ÷ 8:

48 ÷ 8 = 6

48 is divisible by 8.

2. 32 ÷ 5:

32 ÷ 5 = 6 R2

32 is not divisible by 5.

## Divisibility by 1-10

Determining whether a number is divisible by a certain integer by manually dividing and checking for a remainder can get tedious very quickly as the numbers get larger. Fortunately, there are some quick tests that can be used to check whether a given number is divisible by certain integers. In some cases, it may be faster to perform the division, but in others, these tests can save time. Below are divisibility tests for the numbers 1-10.

### Divisibility by 1

All numbers are divisible by 1. No matter what the number is, dividing it by 1 will result in the same number.

### Divisibility by 2

If the digit in the ones place (last digit) of the number is even (0, 2, 4, 6, 8), then the number is divisible by 2.

Example

Test whether the following numbers are divisible by 2.

1. 5568:

8 is divisible by 2, so 5568 is divisible by 2.

2. 527:

7 is not divisible by 2, so 527 is not divisible by 2.

### Divisibility by 3

Find the sum of all of the digits in the number. If the sum of the digits of a number is divisible by 3, then the number is divisible by 3.

Example

Test whether the following numbers are divisible by 3.

1. 273:

2 + 7 + 3 = 12

12 is divisible by 3, so 273 is divisible by 3.

2. 323:

3 + 2 + 3 = 8

8 is not divisible by 3, so 323 is not divisible by 3.

### Divisibility by 4

If the number formed by the last 2 digits of a number is divisible by 4, then the number is divisible by 4.

Example

Test whether the following numbers are divisible by 4.

1. 428:

28 ÷ 4 = 7

28 is divisible by 4, so 428 is divisible by 4.

2. 1055:

55 ÷ 4 = 13 R3

55 is not divisible by 3, so 1055 is not divisible by 3.

### Divisibility by 5

If the last digit in a number is 5 or 0, then the number is divisible by 5.

Example

Test whether the following numbers are divisible by 5.

1. 3325:

The last digit in 3325 is 5, so 3325 is divisible by 5.

2. 325270:

The last digit in 325270 is 0, so 325270 is divisible by 5.

3. 4872:

The last digit in 4872 is neither 0 nor 5, so it is not divisible by 5.

### Divisibility by 6

If a number is divisible by both 2 and 3, then it is divisible by 6.

Example

Test whether the following numbers are divisible by 6.

1. 2358:

2358 ÷ 2 = 1179

2358 ÷ 3 = 786

2358 is divisible by both 2 and 3, so it is divisible by 6. Note that we could have also used the divisibility tests for 2 and 3 rather than completing the division problem; the conclusion would have been the same. 8 is an even number so 2358 is divisible by 2. 2 + 3 + 5 + 8 = 18, which is divisible by 3, so 2358 is divisible by 3.

2. 4528:

4528 ÷ 2 = 2264

4528 ÷ 3 = 1509 R1

4528 is divisible by 2 but not 3, so 4528 is not divisible by 6.

3. 123:

123 ÷ 3 = 41

123 ÷ 2 = 61 R1

123 is divisible by 3 but not by 2, so 123 is not divisible by 6.

### Divisibility by 7

Multiply the last digit in the number by 2 then, excluding the last digit, subtract the product from the original number. If the result is divisible by 7, then the original number is divisible by 7.

Example

Test whether the following numbers are divisible by 7.

1. 567:

7 × 2 = 14

56 - 14 = 42

42 ÷ 7 = 6

42 is divisible by 7, so 567 is divisible by 7.

2. 548:

8 × 2 = 16

54 - 16 = 38

38 ÷ 7 = 5 R3

38 is not divisible by 7, so 548 is not divisible by 7.

### Divisibility by 8

If the last 3 digits in a number are 0, the number is divisible by 8. If the last 3 digits are not 0, but the number formed by the last 3 digits of the original number is divisible by 8, then the original number is divisible by 8.

Example

Test whether the following numbers are divisible by 8.

1. 231968:

968 ÷ 8 = 121

968 is divisible by 8, so 231968 is divisible by 8.

2. 347823000:

The last 3 digits of 347823000 are 0s, so 347823000 is divisible by 8.

### Divisibility by 9

If the sum of the digits of a number is divisible by 9, then the number is divisible by 9. Alternatively, a number that is divisible by 3, twice, is divisible by 9.

Example

Test whether the following numbers are divisible by 9.

1. 2349:

2 + 3 + 4 + 9 = 18

18 is divisible by 9, so 2349 is divisible by 9.

2. 405

405 ÷ 3 = 135

135 ÷ 3 = 45

405 is divisible by 3, twice, so 405 is divisible by 9.

### Divisibility by 10

If the last digit in a number is 0, then the number is divisible by 10.

Example

Test whether the following numbers are divisible by 10.

1. 758290:

The last digit in 758290 is 0, so 758290 is divisible by 10.

There are divisibility tests for other numbers as well, but in some cases the divisibility test can be more tedious than performing the division. Only divisibility tests for 1-10 are shown since they are relatively simple.