A number theory problem by Paola Ramírez

Number Theory Level 2

Find the least positive integer x x such that 13 divides ( x 2 + 1 ) (x^2+1) .


The answer is 5.

Paola Ramírez by Paola Ramírez , 24, Mexico

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2 solutions

Amélie B
Jan 7, 2015

We want x^2 + 1 to be a multiple of 13 and we want x to be the least positive integer possible. The first multiple of 13 is 13 but x^2 + 1 = 13 implies that x = 2sqrt(3) which doesn't satisfy the requirement. The second multiple of 13 is 26 which implies that x=5. Therefore the answer is 5.

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Edwin Gray
Apr 15, 2019

Let x = 13a + b, where b < 13, x^2 + 1 = 169a^2 + 26ab + b^2 + 1. So 13 divides b^2 + 1, where b <13. Trying b = 1,2,3,4,5, we find that 5^2 + 1 = 26 works.

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