Proof of divisibility by 8
Theorem: An integer with at least three digits is divisible by 8 if its last three digits as a number is divisible by 8.
Proof:
For any arbitrary integer anan-1an-2 ....a1a0,
we will show that the integer is divisible by 8 if a2a1a0
is divisible by 8.
The integer can be partially expanded as
anan-1an-2 ....a2a1a0
= (anx10n + an-1x10n-1
+ an-2x10n-2 + .... + a3)x1000
+ a2a1a0
= 8x125(anx10n + an-1x10n-1 + an-2x10n-2 + .... + a3) + a2a1a0
Since the term 8x125(anx10n + an-1x10n-1 + an-2x10n-2 + .... + a3) is divisible by 8, the integer anan-1an-2 ....a1a0 is divisible by 8 if the number a2a1a0 is divisible by 8.