Salute to Powers

Number Theory Level 3

True or false :

$\quad$ $\large 2^{123456789} \times 3^{246913578} - 1$ is divisible by 17.

True False

1 solution

Hana Wehbi
May 13, 2016

For any positive integer $n$ , $2^{n}*3^{2n}-1$ is always divisibile by 17. We can prove that by induction.

Or by modulo arithmetic:
$2^n \times 3^{2n}-1 = 18^n -1 \equiv 1^n-1 = 0 (\text{ mod }17)$

展豪張 - 5 years, 1 month ago

You are right too.

Hana Wehbi - 5 years, 1 month ago

And actually I did not realize the $1:2$ when I was doing it. I used Euler's theorem......

展豪張 - 5 years, 1 month ago