Question : If the 9-digit number 97x4562y8 is divisible by 88, what is the value of $\left(x^2+y^2\right)$ for the smallest value of y, given that x and y are natural numbers?
Option 1: 64
Option 2: 68
Option 3: 76
Option 4: 80
Correct Answer: 80
Solution :
Factors of 88 = 8 × 11
If a number is divisible by 88 then it will be divisible by 8 and 11 also.
The given number is 97x4562y8.
For the smallest possible value of y considering that y is a natural number,
2y8 will be divisible by 8
So, y will be 4 for 248, which is divisible by 8.
As y = 4,
Sum of even places in number = 8 + 2 + 5 + x + 9 = 24 + x
Sum of odd places in number = 7 + 4 + 6 + 4 = 21
Difference between sums = 24 + x – 21 = 3 + x
For 3 + x to be divisible by 11, x = 11 – 3 = 8
Value of $x^2 + y^2$
= $4^2 + 8^2$
= 16 + 64
= 80
Hence, the correct answer is 80.
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