How do you know if a number is divisible by 7 or 8?

Answer 1

A number is divisible by 7, or 8, if when it is divided by 7, or 8, there is no remainder.

To check if a number is so divisible without doing the actual division is easy for 8:

If the last 3 digits of the number are divisible by 8, then so is the whole number.

Alternatively, add the last (units) digit to twice the preceding (tens) digit to four times the preceding (hundreds) digit; if t his sum is divisible by 8 then so is the original number.

To check if a number is divisible by 7 is not easy; often the tests that are given take just as long as (if not longer than) dividing the original number by 7.

One test:

1. Split the original number into groups of 3 digits from the right hand end (just like when putting commas into large numbers to make reading them easier)

2. Starting from the right hand block alternatively subtract and add:

2.1 the last digits of each block

2.2 the middle digits of each block

2.3 the first digits of each block

3. now add the result of step 2.1 to three times the result of step 2.2 to twice the result of step 2.3

4. If this sum is divisible by 7, then so is the original number.

eg 123456789 → 123 456 789

2.1 sum: 9 - 6 + 3 = 6

2.2 sum: 8 - 5 + 2 = 5

2.3 sum: 7 - 4 + 1 = 4

step 3 sum: 6 + 3 x 5 + 2 x 4 = 29

step 4: 29 is not divisible by 7, so 123456789 is not divisible by 7.

eg 654729180 → 654 729 180

2.1 sum: 0 - 9 + 4 = -5

2.2 sum: 8 - 2 + 5 = 11

2.3 sum: 1 - 7 + 6 = 0

step 3 sum: -5 + 3 x 11 + 2 x 0 = 28

step4: 28 is divisible by 7, so 654729180 is divisible by 7.