How would I solve $53^{1069}$ mod 54?

Question

I am doing practice problems for an upcoming exam and am wondering what approach I might take to solve $53^{1069}$ mod 54. 1069 is a prime number, which means I can't factor the exponent.

Some other similar problems that I solved used Fermat's theorem (if N is prime, then $a^{N-1} \equiv$ 1 mod N), but 54 is not prime so I'm not sure if that is correct.

score 5 · Answer 1 · answered Oct 12 '19 at 16:31

5

Hint: The problem is $(-1)^{1069}\mod 54$ and $1069$ is odd.

answered Oct 12 '19 at 16:31

Wuestenfux

20,964

score 1 · Answer 2 · answered Oct 12 '19 at 16:34

1

$53 \equiv -1 \bmod 54$ thus we can have $53^{1069} \equiv (-1)^{1069} \equiv -1 \equiv 53 \bmod 54$.

answered Oct 12 '19 at 16:34

Lily

86

How would I solve $53^{1069}$ mod 54?

2 Answers2