Euler would have done it easily...

Number Theory Level 3

Find the remainder when $87^{ 554 }$ is divided by $987$ .

The answer is 660.

1 solution

Vighnesh Raut
Jan 9, 2015

$We\quad know\quad that\quad \phi (987)=552\\ So,\quad \quad { 87 }^{ 552 }\equiv \quad 1\quad \quad \quad \quad \quad \quad \quad (mod\quad 987)\\ Hence,{ 87 }^{ 554 }\equiv \quad { 87 }^{ 552 }.{ 87 }^{ 2 }\quad\quad\quad \quad (mod\quad 987)\\ \quad \quad\quad\quad\quad \equiv \quad (1)\quad .7569\quad\quad\quad (mod\quad 987)\\ \quad \quad\quad\quad\quad \equiv \quad 660\quad \quad \quad \quad \quad \quad (mod\quad 987)\quad$

Euler would have done it easily...

The answer is 660.

1 solution

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