Try to Solve Without a Calculator 2

Number Theory Level 2

104 5 2089 − 104 4 2089 ≡ x ( m o d 2089 ) 1045^{2089}-1044^{2089}\equiv x \pmod{2089} What is the minimum value of x x , without using a calculator?

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Drex Beckman by Drex Beckman , 23, USA

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1 solution

Department 8
Jan 14, 2016

Since 2089 2089 is prime and we know a p ≡ a ( m o d p ) a^{p} \equiv a \pmod p for prime p p . Now the given question can be rewritten as

104 5 2089 − 104 4 2089 ≡ 1045 − 1044 = 1 ( m o d 2089 ) 1045^{2089}-1044^{2089} \equiv 1045 - 1044=\boxed{1} \pmod{2089}

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Nice solution!

Drex Beckman - 5 years, 5 months ago

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