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Algebra Level 4

Let p p be an integer such that both roots of the equation 5 x 2 − 5 p x + ( 66 p − 1 ) = 0 5x^2 - 5px + (66p - 1) = 0 are positive integers. Find the value of p.


The answer is 76.

Dev Sharma by Dev Sharma , 20, India

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1 solution

Dev Sharma
Oct 13, 2015

The discriminant must be a perfect square number implies that

( 5 p ) 2 − 20 ( 66 p − 1 ) = n 2 (5p)^{2} - 20(66p - 1) = n^2 , so solving this equation gives

( 5 p − 132 ) 2 − n 2 = 17404 (5p - 132)^{2} - n^2 = 17404 .

Note that 5|n so p − 1 = 0 ( m o d 5 ) p-1=0 (mod5) Take p = 5k + 1 and put in the equation and comparing gives p = 76

3 Helpful 1 Interesting 0 Brilliant 0 Confused

Why should it be a perfect square?

Marvel Wijaya - 5 years, 7 months ago

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read Discriminant

Dev Sharma - 5 years, 7 months ago

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