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.
Answers
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:
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