*by Eugene Chen of the Amador Valley Math Team*Divisibility rules are very important to math competitions. Instead of
performing lots of long division to find prime factors, we can use
divisibility rules to narrow down our search. In most math competitions, you will
usually need to know divisibility rules up to

. This lesson includes divisibility rules up to

.

An integer is divisible by

if and only if the last digit of it is

,

,

,

, or

. For example,

is divisible by

because its last digit is

, but

is not divisible by

because its last digit is

.

An integer is divisible by

if and only if the sum of its digits is divisible by

. For example,

is divisible by

because the sum of its digits is

, which is divisible by

, but

is not divisible by

because the sum of its digits is

, which is not divisible by

.

An integer is divisible by

if the number formed by the last two digits of it is divisible by

. For example,

is divisible by

, because the number formed by the last two digits of it (

) is divisible by

, but

is not divisible by

, because the number formed by the last two digits of it (

) is not divisible by

.

An integer is divisible by

if its last digit is

or

. For example,

is divisible by

because it ends in

, but

is not divisible by

because it ends in

.

An integer is divisible by

if it satisfies the divisibility rules for both

and

.

To determine if an integer is divisible by

,
take the last digit, double it, and subtract it from the number formed
by the remaining digit. Continue this process until you reach a number
that you know is or is not divisible by

. If it is divisible by

, the original number is. If it isn't divisible by

, then the original number is not. For example,

is divisible by

because doubling the last digit and subtracting from the remaining digits yields

, which is divisible by

, but

is not divisible by

because doubling the last digit and subtracting from the remaining digits yields

, which is not divisible by

.

An integer is divisible by

if the number formed by its last three digits is divisible by

. For example,

is divisible by

because

is, but

is not divisible by

because

is not divisible by

.

An integer is divisible by

if the sum of its digits is divisible by

. For example,

is divisible by

because the sum of its digits is

, which is divisible by

, but

is not divisible by

because the sum of its digits is

, which is not divisible by

.

An integer is divisible by

if its units digit is

. For example,

is divisible by

because it ends in a

, but

is not divisible by

because it ends in

.

To determine if an integer is divisible by

, alternately add and subtract the digits from left to right. If the resulting integer is divisible by

, the original number is. For example,

is divisible by

, because alternately adding and subtracting its digits yields

, which is divisible by

, but

is not divisible by

, because alternately adding and subtracting its digits yields

, which is not divisible by

.

An integer is divisible by

if it satisfies the divisibility rules for

and

.

To determine if an integer is divisible by

, remove the last digit, multiply it by

,
and subtract this from the remaining digits. Continue this until you
reach a number that you know is divisible or not divisible by

. If it is divisible by

, then the original number is divisible by

, but if it is not divisible by

, the original number is not. For example,

is divisible by

, because multiplying the last digit by

and subtracting it from the rest of the number yields

, which is divisible by

, but

is not divisible by

, because multiplying the last digit by

and subtracting it from the rest of the number yields

, which is not divisible by

.

Exercise 1: Determine if the following numbers are divisible by

:

a)
b)
c)
d)
Problem 1: Prove the divisibility rules for

and

.

Problem 2: Prove the divisibility rules for

,

, and

.

Problem 1 Hint: Break up the number into two parts.

Problem 2 Hint: Prove that the operation in the divisibility rules does not change the modular residue.