Remainder!

Algebra Level 3

Find the remainder when x 1999 x^{1999} is divided by x 2 − 1 x^2 - 1 .

− 2 x -2x 2 x 2x x x 5 x 5x

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Dev Sharma by Dev Sharma , 20, India

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1 solution

Otto Bretscher
Apr 7, 2016

x 1999 = ( x 2 ) 999 x ≡ x ( m o d x 2 − 1 ) x^{1999}=(x^2)^{999}x\equiv \boxed{x} \pmod {x^2-1}

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Is it x 2 = 1 ( m o d ( x 2 − 1 ) ) x^2 = 1 (mod(x^2 - 1)) ?

Dev Sharma - 5 years, 2 months ago

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Yes! That means, by definition of modular arithmetic, that x 2 − 1 x^2-1 is divisible by x 2 − 1 x^2-1 .

Otto Bretscher - 5 years, 2 months ago

Seems like number theory problem

Resha Dwika Hefni Al-Fahsi - 5 years, 2 months ago

Of course.

Kushagra Sahni - 5 years, 2 months ago

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