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You are given
x + 125 = n^2 (1)
X + 176 = m^2 (2)
Subtract (1) from (2). You will get
m^2 - n^2 = 176 - 125, or
m^2 - n^2 = 51.
It implies
(m+n)*(m-n) = 51. (*)
It is my basic equation, and I am going to find all its integer solutions.
Notice, that if the pair of positive numbers (m,n) is the solution, then 3 other pairs
(-m,-n), (-m,n) and (m,-n) are the solutions, too.
Therefore, I will look and search for positive solution pairs (m,n) only, keeping in mind that
every such a pair brings 3 other solutions, playing with their signs.
But since my "x" depends only on m^2 and n^2, this playing with signs does not matter, at all.
Since "m" and "n" are integer numbers, it implies that
EITHER
m + n = 51, (3)
m - n = 1, (4)
OR
m + n = 17, (5)
m - n = 3. (6)
From equations (3), (4), the solution is m= 26, n= 25.
From equations (5), (6), the solution is m= 10, n= 7.
Thus EITHER
x + 125 = n^2 = 25^2 = 625, which implies x= 625-125 = 500,
OR
x + 125 = n^2 = 7^2 = 49, which implies x= 49-125 = -76.
ANSWER. The problem has two solutions . The number under the question is EITHER 500 OR -76.
Solved.
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It is how this problem is expected to be solved and should be solved.
