Proofathon Problem 1

Number Theory Level 5

Determine the remainder when $6000^{5999^{\dots^{1}}}$ is divided by $2013$ .

The answer is 1974.

1 solution

Muhammad Ihsan
May 11, 2015

${ 6000 }^{ { 5999 }^{ ...1 } }\quad mod\quad 3\quad =\quad 0\\ .\\ { 6000 }^{ { 5999 }^{ { 5998 }^{ { 5997 }^{ ...1 }\quad mod\quad 2 }\quad mod\quad 4 }\quad mod\quad 10 }\quad mod\quad 11\quad =\quad 5\\ .\\ { 6000 }^{ { 5999 }^{ { 5998 }^{ { 5997 }^{ ...1 }\quad mod\quad 8 }\quad mod\quad 16 }\quad mod\quad 60 }\quad mod\quad 61\quad =\quad 11\\ .\\ x\quad =\quad 0\quad mod\quad 3\\ x\quad =\quad 5\quad mod\quad 11\\ x\quad =\quad 11\quad mod\quad 61\\ .\\ x\quad =\quad \boxed { 1974 } \quad mod\quad 2013$

Why is the third modulus mod 13 instead of mod 61? The number 2013 is 3 * 11 * 61.

John Barczynski - 6 years ago

thanks for your correction

Muhammad Ihsan - 6 years ago

Proofathon Problem 1

The answer is 1974.

1 solution

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