A.4 Divisibility rules

Divisibility rules

Key Notes:

What are Divisibility Rules?

Divisibility rules are shortcuts to help you figure out if a number can be divided evenly by another number without actually doing the division. They’re super useful for simplifying fractions, solving problems, and understanding numbers better! Let’s explore some common divisibility rules.

Divisibility by 2

Rule: A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.
Example:
24 is divisible by 2 because it ends in 4.
12 is divisible by 2 because its last digit is 2.
346 is divisible by 2 because its last digit is 6.
1000 is divisible by 2 because its last digit is 0.
23 is not divisible by 2 because its last digit is 3.

Divisibility by 3

Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
Example:
123 is divisible by 3 because 1 + 2 + 3 = 6, and 6 is divisible by 3.
456 is divisible by 3 because 4 + 5 + 6 = 15, and 15 is divisible by 3.
78 is divisible by 3 because 7 + 8 = 15, and 15 is divisible by 3.
100 is not divisible by 3 because 1 + 0 + 0 = 1, and 1 is not divisible by 3.

Divisibility by 4

Rule: A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
Example:
For 312, the last two digits are 12, and 12 is divisible by 4.
112 is divisible by 4 because 12 is divisible by 4.
324 is divisible by 4 because 24 is divisible by 4.
1600 is divisible by 4 because 00 is divisible by 4.
210 is not divisible by 4 because 10 is not divisible by 4.

Divisibility by 5

Rule: A number is divisible by 5 if its last digit is 0 or 5.
Example:
45 is divisible by 5 because it ends in 5.
25 is divisible by 5 because its last digit is 5.
130 is divisible by 5 because its last digit is 0.
1000 is divisible by 5 because its last digit is 0.
12 is not divisible by 5 because its last digit is 2.

Divisibility by 6

Rule: A number is divisible by 6 if it is divisible by both 2 and 3.
Example:
18 is divisible by 6 because it’s even (divisible by 2) and the sum of digits (1+8=9) is divisible by 3.
24 is divisible by 6 because it’s even (divisible by 2) and 2 + 4 = 6 (divisible by 3).
132 is divisible by 6 because it’s even (divisible by 2) and 1 + 3 + 2 = 6 (divisible by 3).
12 is divisible by 6 because it’s even (divisible by 2) and 1 + 2 = 3 (divisible by 3).
25 is not divisible by 6 because although it is not divisible by 2, it is not divisible by 3.

Divisibility by 7

Rule: Double the last digit and subtract it from the rest of the number. If the result is divisible by 7 (including 0), then the original number is divisible by 7.
Example: For 203, double the last digit (6), subtract from the rest (20 – 6 = 14), and since 14 is divisible by 7, so is 203.

Divisibility by 8

Rule: A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
Example: For 1,856, the last three digits are 856, and 856 is divisible by 8.

Divisibility by 9

Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
Example:
For 729, the sum of the digits is 7 + 2 + 9 = 18, and 18 is divisible by 9.
81 is divisible by 9 because 8 + 1 = 9, and 9 is divisible by 9.
279 is divisible by 9 because 2 + 7 + 9 = 18, and 18 is divisible by 9.
999 is divisible by 9 because 9 + 9 + 9 = 27, and 27 is divisible by 9.
100 is not divisible by 9 because 1 + 0 + 0 = 1, and 1 is not divisible by 9.

Divisibility by 10

Rule: A number is divisible by 10 if its last digit is 0.
Example:
50 is divisible by 10 because its last digit is 0.
230 is divisible by 10 because its last digit is 0.
1000 is divisible by 10 because its last digit is 0.
45 is not divisible by 10 because its last digit is 5.

Divisibility by 11

Rule: A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is divisible by 11.
Example: For 2728, sum of digits in odd positions (2 + 2 = 4) and sum of digits in even positions (7 + 8 = 15), difference is |4 – 15| = 11, which is divisible by 11.

Divisibility by 12

Rule: A number is divisible by 12 if it is divisible by both 3 and 4.
Example: For 144, since 144 is divisible by 3 (sum of digits is 9) and 4 (last two digits 44 are divisible by 4), it is divisible by 12.

Tips for Students

Practice: Use these rules to check your work and solve problems.
Patterns: Look for patterns in the rules to help remember them.
Examples: Work through several examples to see how the rules work.
2: Last digit is even (0, 2, 4, 6, 8).
3: Sum of digits is divisible by 3.
4: The number formed by last 2 digits is divisible by 4
5: Last digit is 0 or 5.
6: Divisible by both 2 and 3.
9: Sum of digits is divisible by 9.
10: Last digit is 0.

Learn with an example

😀 Is 84,410,298 divisible by 5?

Try the “divisible by 5” rule on 84,410,298.

Look at the ones digit:

84,410,298

The ones digit is 8.

The rule says that 84,410,298 is not divisible by 5.

😀 Is 37,770,710 divisible by 10?

Try the “divisible by 10” rule on 37,770,710.

Look at the ones digit:

37,770,710

The ones digit is 0.

The rule says that 37,770,710 is divisible by 10.

😀 Is 8,611,390 divisible by 2?

Try the “divisible by 2” rule on 8,611,390.

Look at the ones digit:

8,611,390

The ones digit is 0.

The rule says that 8,611,390 is divisible by 2.

let’s practice!