Question

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

Answer 1

I + 200 = A2 : 200 added to I (unknown integer) gives a square.

I + 276 = B2 : 276 added to I (unknown integer) gives another square.

B2 = A2 + 76 : eliminate I from the two equations.

add squares A2 (0, 1, 4, 9, 16, 25,...) to 76 till you obtain another square B2.

76 + 182 = 400 = 202

A2 = 182 and B2 = 202

I = A2 - 200 = 124

Answer 2

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let the perfect square is p^2 and q^2

by the question,  200+i = p^2 and 276+i = q^2

 subtract them, 76 = q^2 - p^2

                       => (p+q)(q-p) = 76

  the factor of 76 is 1,2,4,19,38 and 76.

                     so here we use 38 and 2

        compare this => p+q = 38 and q-p = 2

                               so, p = 18 and q = 20

 this implies that  ---> 200 + i = 324 => i = 124  Ans....                

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