Large Powers!

Algebra Level 4

P=2008^2007-2008;

Q=2008²+2009.

The remainder when P is divided by Q is..


The answer is 4032066.

Naitik Sanghavi by naitik sanghavi , 20, India

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1 solution

Surya Prakash
Aug 5, 2015

Let x = 2008 x = 2008 .

So,

Q = x 2 + x + 1 Q = x^{2} + x +1 x 3 ≡ 1 m o d Q x^{3} \equiv 1 \mod Q x 2007 ≡ 1 m o d Q x^{2007} \equiv 1 \mod Q x 2007 − x ≡ 1 − x ≡ 1 − x + x 2 + x + 1 ≡ x 2 + 2 ≡ 4032066 m o d Q x^{2007} -x \equiv 1-x \equiv 1-x+x^{2} + x + 1 \equiv x^{2} + 2 \equiv 4032066 \mod Q

Therefore, P ≡ 4032066 m o d Q P \equiv \boxed{4032066} \mod Q .

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Moderator note:

As a side note, you still have to check that 4032066 < 200 8 2 + 2009 4032066 < 2008^2 + 2009 . This would guarantee that you found the remainder, instead of it's modulo equivalence.

Challenge Master : It's Obvious. Since x 2 + 2 < x 2 + x + 1 x^{2} + 2 < x^{2} + x + 1 , since x = 2008 x = 2008 .

Surya Prakash - 5 years, 10 months ago

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