Find the remainder when $9^{16} - 5^{16}$ is divided by $14$.

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Solution 1

Hint:

Since

$$9 \equiv -5 \pmod{14}$$

$$9^{16} - 5^{16} \equiv (-5)^{16} - 5^{16} \pmod{14}$$

Solution 2

Write $$ 9^{16}-5^{16}=(9^8-5^8)(9^8+5^8)\tag{1}. $$ 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 $$ 9^8-5^8\equiv 9^2-5^2\equiv4(14)\equiv0\mod{14} $$ and the result follows from (1).

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Piyush Sevaldasani
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Piyush Sevaldasani

Updated on December 22, 2022

Comments

  • Piyush Sevaldasani
    Piyush Sevaldasani 11 months

    Tried a lot. Though unable to find starting point.

    • Angelo Rendina
      Angelo Rendina over 6 years
      Hint: $x^2-y^2=(x+y)(x-y)$