Question

Prove that for every positive integer n, $1^n+8^n-3^n-6^n$ is divisible by 10.

Answer 1

Since 10 is the product of two primes 2 and 5, it will suffice to show that the given expression is divisible both by 2 and 5. To do so, we shall use the simple fact that if a and b be any positive integers, then $a^n-b^n$ is always divisible by $a-b$.
Writting $A\equiv 1^n+8^n-3^n-6^n$
$=(8^n-3^n)-(6^n-1^n)$
we find that $8^n-3^n$ and $6^n-1^n$ are both divisible by 5, and consequently A is by $5(=8-3=6-1)$. Again, writing $(8^n-6^n)-(3^n-1^n)$, we find that A is divisible by 2($=8-6=3-1$). Hence A is divisible by 10.