Squares are Awesome

Number Theory Level 1

If 200 is added to a positive integer R, the result is a square number. If 276 is added to to the same integer R, another square number is obtained. Find R.


The answer is 124.

Vighnesh Raut by Vighnesh Raut , 23, India

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4 solutions

Vighnesh Raut
Apr 7, 2014

R + 200 = square number A and R + 276 also equal to a square number B. This gives us that A=B+76 . By trying some squares (1,4,9,16,....) , we get B= 324 and A = 400. Therefore, R = 324 - 200 = 400 - 276 = 124.

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Oanh Nguyễn
Apr 16, 2014

R + 200 = X^2 R + 276 = Y^2 so we have Y^2 - X^2 = 76 which means (X + Y)(Y - X) = 76. Because 76 = 1X76 = 2X38 = 4X19. We get X = 20 and Y = 18

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you did not state how you got 20 and 18.

Blair Mallory - 4 years, 10 months ago
Mehbuba Hossain
Apr 23, 2014

R+200=a & R+276=b,where a&b are a square no. Which implies b=a+76 & obviously a,b>=289. So by trying some squares(289,324,361,400) we get that,b=400 & a=324. So R=124 a solution.

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Harshad Kale
Apr 19, 2014

324 AND 400 BOTH ARE SQUARES AD 200+124=324 AND 276+124=400

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