You may also like

problem icon


What is the smallest perfect square that ends with the four digits 9009?

problem icon

More Mods

What is the units digit for the number 123^(456) ?

problem icon

Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?


Age 14 to 18 Challenge Level:

Murat from Turkey sent the following solution.

If $g(n)=(1+8^n-3^n)$ is divisible by 6, then $(1+8^n-3^n-6^n)$ is also divisble by 6. It can be verified that


for all positive integers $n> 3$. Since $g(1)$ and $g(3)$ are divisible by 6, it follows that $g(5)$ is also. By induction, it can be shown that for all odd $n$, $g(n)$ is divisible by 6. Since $g(2)$ and $g(4)$ are not divisible by 6, this is not the case for even $n$.

An alternative approach is to use the facts that powers of odd numbers are always odd; powers of even numbers are always even; also the difference of two odd numbers is even. Hence $N=1^n+8^n-3^n-6^n$ is even (odd + even - odd - even).

It remains to decide whether or not $N$ is divisible by 3.

$N \equiv 1 + (-1)^n - 0 - 0$ (mod 3)

This shows that $N \equiv 0$ (mod 3) if $n$ is odd and hence $N$ will be divisible by 6 for all odd values of $n$. However, $N \equiv 2$ (mod 3) if $n$ is even and so $N$ cannot be divisible by 6 for even values of $n$.