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Number Theory Level 3

Compute the remainder obtained when 9 1990 9^{1990} is divided by 11.


The answer is 1.

Krishna Ar by Krishna Ar , 21, India

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1 solution

Daniel Liu
Jun 27, 2014

Note that a 10 ≡ 1 ( m o d 11 ) a^{10}\equiv 1\pmod{11} as long as ( a , 11 ) ≠ 0 (a,11)\ne 0 by Fermat's Little Theorem. Thus, 9 1990 = ( 9 199 ) 10 ≡ 1 ( m o d 11 ) 9^{1990}=(9^{199})^{10}\equiv \boxed{1}\pmod{11}

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Yep did it same way. A little bit faster than @mathh mathh

Mardokay Mosazghi - 6 years, 11 months ago

Nice solution. I just observed that 3 5 3^{5} is congruent to 1 (mod 11), and thus so is 3 3980 3^{3980} .

Brian Charlesworth - 6 years, 11 months ago

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