Perfect Squares Problem

Number Theory Level 1

Let $n$ be a positive integer. If $n + 100$ is a perfect square and $n + 168$ is also a perfect square, then find $n$ .

The answer is 156.

2 solutions

Edwin Gray
Aug 24, 2018

Let n + 100 = k^2, n + 168 = m^2. Subtracting, 68 = m^2 - k^2 = (m - k)(m + k). Since both factors must be even, m - k = 2, m+ k = 34. Adding, m = 18, k = 16. Substituting in either of the original equations, n = 156. Ed Gray

Charlie Feinson
Dec 6, 2017

n + 100 = y^2

n + 168 = (y + a)^2

y^2 + 68 = y^2 + 2ay + a^2

68 = 2ay + a^2

a = 2, y = 16.

n + 100 = 16^2 = 156

n = 156

(n + 168 = 156 + 168 = 324 = 18^2)

you can use latex .see @Munem Sahariar guide

Saksham Jain - 3 years, 6 months ago

Perfect Squares Problem

The answer is 156.

2 solutions

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