DIVISIBILITY RULES

Let x and y be integers. When we say "x is divisible by y" we mean that when x is divided by y there will be no remainder. Following are divisibility rules for common divisors.

Divisibility rules are helpful for example, when we wish to reduce fractions. For example, we know that can be reduced because both 46 and 48 can be divided by 2.

Divisibility Rule for 2

A number is divisible by 2 if it is even (that is, if the last digit is either 0, 2, 4, 6, or 8.)

Example: 28 is divisible by 2, 11 is not divisible by 2.

Divisibility Rule for 5

A number is divisible by 5 if the last digit is 0 or 5.

Example: 15 is divisible by 5, 23 is not divisible by 5.

Divisibility Rule for 10

A number is divisible by 10 if the last digit is a zero.

Example: 100,000 is divisible by 10.

Divisibility Rule for 4

A number is divisible by 4 if the number formed by the last two digits of the number is divisible by 4.

Example: 13,524 is divisible by 4 since 24 is divisible by 4.

Divisibility Rule for 3

A number is divisible by 3 if the sum of the digits is divisible by 3.

Example: 85,032 is divisible by 3 since the sum of the digits, 18, is divisible by 3. (Note that 8+5+0+3+2=18.)

Divisibility Rule for 9

A number is divisible by 9 if the sum of the digits is divisible by 9.

Example: 261 is divisible by 9 since the sum of the digits, 9, is divisible by 9. (Note that 2+6+1=9.)

Example: Is 9801 prime or composite?

Solution: We can divide 9801 by the prime numbers, 2, 3, 5, 7, 11, … until we find a factor of 9801. However, if we add the digits of 9801 (9+8+0+1=18), we see that their sum is divisible by both 3 and 9. Therefore, 9801 is composite.