A month or so ago, when I started learning exploring Number Theory, I looked more deeply into divisibility rules. The one that I got from Google for 8 was "A number is divisible by 8 if and only if its last three digits are divisible by 8".

This method is obviously accurate and can be proved really easily. However, it's not that useful... When I see a 3-digit number, I can not tell immediately if it is a multiple of 8. So I thought about how I can verify if 3-digit number is a multiple of 8 very quickly without a calculator, and came up with this.

A number \(\overline{...abc}\), (meaning a number ending in the three digits a, b and c) is a multiple of 8 if...

(1) \(a\) is even and \(\overline{bc}\) is a multiple of 8

OR

(2) \(a\) is odd and \(\overline{bc}\) is a multiple of 4 and not 8.

This method is much simpler than the one I found from Google, because you need only remember 2-digit multiples of 4 and 8 instead of 3-digit multiples.

I wrote this proof for my method. It's my first attempt at writing a proof, so my format probably won't be very nice, but the mathematics is valid.

Thanks for reading!

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