Answer to Question #275904 in Combinatorics | Number Theory for Vishnu Kumar

Question

Prove that 3n+4n+5n is divisible by 12 whenever n is an odd positive integer.

Answer 1

As i understand it should be $3^n+4^n+5^n$

To prove that number is divisible by 12 we will prove that it is divisible by both 3 and 4

Divisible by 3:

$3^n$ is obviously divisible by 3 for every odd integer n, then the point is to prove that $4^n+5^n$ is divisible by 3

According to abbreviated multiplication formula for odd degrees:

$4^n+5^n=(4+5)(4^{n-1}-4^{n-2}*5+4^{n-3}*5^2-...-4*5^{n-2}+5^{n-1})=9*(4^{n-1}-4^{n-2}*5+4^{n-3}*5^2-...-4*5^{n-2}+5^{n-1})⋮3$

Divisible by 4:

$4^n$ is obviously divisible by 4 for every odd integer n, then the point is to prove that $3^n+5^n$ is divisible by 4

According to abbreviated multiplication formula for odd degrees:

$3^n+5^n=(3+5)(3^{n-1}-3^{n-2}*5+3^{n-3}*5^2-...-3*5^{n-2}+5^{n-1})=8*(3^{n-1}-3^{n-2}*5+3^{n-3}*5^2-...-3*5^{n-2}+5^{n-1})⋮4$

The statement has been proven