Find the remainder when $9^{16}  5^{16}$ is divided by $14$.
1,132
Solution 1
Hint:
Since
$$9 \equiv 5 \pmod{14}$$
$$9^{16}  5^{16} \equiv (5)^{16}  5^{16} \pmod{14}$$
Solution 2
Write $$ 9^{16}5^{16}=(9^85^8)(9^8+5^8)\tag{1}. $$ By Euler's theorem, $$ 9^6\equiv 1\mod 14;\quad 5^6\equiv 1\mod 14 $$ since $5$ and $9$ are coprime to $14$. Then $$ 9^85^8\equiv 9^25^2\equiv4(14)\equiv0\mod{14} $$ and the result follows from (1).
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Author by
Piyush Sevaldasani
Updated on December 22, 2022Comments

Piyush Sevaldasani 11 months
Tried a lot. Though unable to find starting point.

Angelo Rendina over 6 yearsHint: $x^2y^2=(x+y)(xy)$
