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

Answer:

124

Step-by-step explanation:

Given :

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. ​

Solution :

We know that,

Sum of n consecutive odd numbers = n²,.

I.e.,

If n = 2,

⇒ 1 + 3 = 4 = 2²

If n = 3,

⇒ 1 + 3 + 5 = 9 = 3²,.

Hence,.

(I + 200) + some odd numbers = I + 276

Let (I + 200) = a²

Then, the odd numbers be (a - 2), a , (a + 2) , (a + 4)

⇒ I + 200 + (a - 2) + a + (a + 2) + (a + 4) = I + 276

⇒ 200 + 4a + 4 = 276

⇒ 4a = 72

⇒ a = 18,.

⇒ I + 200 = 18²

⇒ I + 200 = 324

⇒ I = 324 - 200 = 124