Proof of Divisibility by 8

If you have read the discussion on divisibility by 4, the proof for divisibility by 8 is somewhat similar. To know that the number is divisible by 4, we just have to look at the last two digits. In divisibility by 8, we look at the last three digits. For example, the number 10938648 is divisible by 8 because the last three digits which is 648 is divisible by 8.

Notice that number with digits abcd can be expanded as

1000a + 100b + 10c + d

or

10^3a + 10^2b + 10c + d.

Now, since 1000 is divisible by 8, any multiples of 1000 are divisible by 8. Since any integer can be represented as

10^na_n + 10^{n-1}a_{n-1} + 10^{n - 2}a_{n-2} + \cdots + 10^3a_3 + 10^2a_2 + + 10^1a_1 + 10^0a_0.

Don’t be intimidated, this is just a generalization of the decimal expansion above.

Now, all integral powers of 10 that are equal to 3 or greater are multiples of 1000, so, we only have to check

10^2a_2 + 10^1a_1 + 10^0a_0

which is the last three digits of any positive integer.

That is the reason, why we only need to check the three digits of any integer and see if it is divisible by 8.

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