# Divisibility Test: 2, 4, and 8

The divisibility rules for 2, 4 and 8 are simple and similar.

A number is divisible by 2 if its last digit is divisible by 2.

Since 10 is divisible by 2, the divisibility of a number by 2 depends only on the unit’s digit. So all even numbers, i.e. those numbers ending in 0, 2, 4, 6 and 8 are divisible by 2.

A number is divisible by 4 if its last 2 digits are divisible by 4.

100 is evenly divisible by 4 so only the last two digits determines the number’s divisibility by 4. First the number must be even to be divisible by 2.  All odd numbers are not.

A number is divisible by 8 if its last 3 digits are divisible by 8.

1000 is evenly divisible by 8, so we only have to determine if the last 3 digits are divisible by 8.

There are several handy methods to simplify the division of the last digits by 4 or 8:

a) Divide the last 2 digits by 4 or the last 3 digits by 8 using the method in MSC 12.

b) Divide the last two digits by 2, if the quotient is an even number, the number is divisible by 4. Divide the last three digits by 2 twice. If the quotient is an even number, the number is divisible by 8.

c) Add the ultimate (last) digit to twice the penultimate (second to the last) digit and if the sum is divisible by 4, the number is divisible by 4. For 8, add twice the hundreds digit and four times the tens digit to the unit’s digit. If the total is divisible by 8, the number is divisible by 8. This method is not recommended but is instructive in understanding divisibility of 3, 9, and 11.

d) Add or subtract from the digits to arrive at numbers obviously divisible by the number tested.

d.1) Add or subtract 4 or 8 from the last digit to make it zero and if the resulting ten’s digit is an even number, the number is divisible by 4.

Examples:   52 + 8 = 60, 52 is divisible by 4; but 74 – 4 = 70, so 70 is not divisible by 4.

d.2) To find out if a number is divisible by 8, add or subtract 8 or 16 from the last three digits to make the last digit zero and see if the number resulting is divisible by four.  Add or subtract 40 or 80 to make the ten’s digit also zero if needed.  If the hundreds digit becomes even, the number is divisible by 8.

392 + 8 = 400; 392 is divisible by 8;             296 – 16 = 280;  280 – 80 = 200;  296 is divisible by 8

324 + 16= 340;  340-40 = 300; 324 is not divisible by 8.