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For how many positive integers $n$ less than $100$ is $5^n+8^{n+1}+13^{n+2}$ a multiple of $6$?

 Oct 8, 2022
 #1
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The answer is 45.

 Oct 8, 2022
 #2
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5^n + 8^(n+1) + 13^(n + 2)  = (-1)^n + 2^(n+1) + 1  (mod 6)

 

Case 1: n is odd

 

2^(n + 1) = 0 (mod 6)

No values of n will satisfy this as we will need to get a factor of 3 in order to make this expression divisible by 6. 

 

 

Case 2: n is even

 

2^(n+1) + 2 = 0 (mod 6)

 

2^(n+1) = 4 (mod 6)

 

2

4 check

16 check

32 
64 check
128

256 check

 

It seems like all the even powers of 2 leave a remainder of 4 when divided by 6.

However, if we subtract one, we would get an odd number for n which goes against the case restrictions. Therefore, there are 0?

Please correct me if I am wrong.

 Oct 9, 2022

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