Divisibility rules

key notes :

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.

Divisibility by 3

• Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Example: For 123, the sum of the digits is 1 + 2 + 3 = 6, which is 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.

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.

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.

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.

Divisibility by 10

• Rule: A number is divisible by 10 if its last digit is 0.
• Example: 50 is divisible by 10 because it ends in 0.

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.

Learn with an example

let’s practice!