Basic

Number Theory Level 3

What will be the remainder if the number 7 2012 7^{2012} is divided by 25?


The answer is 1.

Kartik Agarwal by Kartik Agarwal , 20, India

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2 solutions

Arulx Z
Jan 11, 2016

7 2012 ≡ 49 1006 ≡ ( − 1 ) 1006 ≡ 1 ( mod 25 ) { 7 }^{ 2012 }\equiv { 49 }^{ 1006 }\equiv { \left( -1 \right) }^{ 1006 }\equiv 1\quad \left(\text{mod}\quad 25 \right)

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Good observation of using 7 2 ≡ − 1 7^2 \equiv -1 . Sometimes we don't have to use Euler's theorem to evaluate such large powers.

Kunwar Moorjani
Jan 10, 2016

(50-1)^2012 (-1)^2012=1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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