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Question

Prove that for every integer n, the number En = 5^n + 2*3^(n-1) +1 is a multiple of 8.

I tried to use induction, but I was told that a more simple way of doing it was using the rules of divisibility by 8.

Any ideas?

Don't use any sort of number theory (modulus). Apparently the answer is very simple and it is only using the rules of when a number is divisible by 8.

I know that the last three digits of any number divisible by 8 is always divisible by 8, but would this be applied here?

Laura C · Accepted Answer

First rewrite 5^n as 5*5^(n-1) so you get
En=5*5^(n-1) + 2*3^(n-1) + 1

Next, bring them to a common denominator, in this case 5^(n-1)*3^(n-1) =>
En=[5*5^(n-1)*3^(n-1) + 2*3^(n-1)*5^(n-1) + 5^(n-1)*3^(n-1)]/[5^(n-1)*3^(n-1)]

Now force 5^(n-1)*3^(n-1) as a common factor so you get
En={[5^(n-1)*3^(n-1)] *(5+2+1)}/[5^(n-1)*3^(n-1)]

Since 5+2+1=8, it proves that En is a multiple of 8

JOS J · Answer

(1 + 2 3^(-1 + n) + 5^n)/8 

Set n from  1 to 10

{1, 4, 18, 85, 411, 2014, 9948, 49375, 245781, 1225624}

Math Question... 10 Points To First Correct Answerer?

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