SOLUTION: Give proof that 9^n-7^n is divisable by 32 if n is an even number

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Question 956942: Give proof that 9^n-7^n is divisable by 32 if n is an even number
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let's look at the general case,
a%5E2-b%5E2=%28a%2Bb%29%28a-b%29
a%5E4-b%5E4=%28a%2Bb%29%28a-b%29%28a%5E2%2Bb%5E2%29
a%5E6-b%5E6=%28a%2Bb%29%28a-b%29%28a%5E2-ab%2Bb%5E2%29%28a%5E2%2Bab%2Bb%5E2%29
a%5E8-b%5E8=%28a%2Bb%29%28a-b%29%28a%5E4%2Bb%5E4%29

As you see from these 5 cases, a-b and a%2Bb are always factors for even powers of n.
In this case,
a-b=9-7=2
a%2Bb=9%2B7=16
So the product 32 is always a factor that why it's always divisible by 32.