SOLUTION: Prove that 3 divides 20^2n − 1 for all n = 0, 1, 2, 3, . . ..

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Question 1020241: Prove that 3 divides 20^2n − 1 for all n = 0, 1, 2, 3, . . ..
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
20%5E%282n%29-1=400%5En-1
When n=0,
400%5E%28m%29-1=1-1=0 divisible by 3.
When n=1,
400%5E%28n%29-1=400-1=399 divisible by 3 since it ends in 9.
When n=2,
400%5E%28n%29-1=160000-1=159999 divisible by 3 since it ends in 9.
As you see the number is a power of 400 so the last digit will always be 0.
Subtracting 1 makes it the last digit a 9 so it will always be divisible by 3.