If “62x2365y” is an eight-digit number divisible by 72, then find the value of ‘x^2 + y^2 – xy’. Options 40 36 32 28

To determine the value of $x^2 + y^2 - xy$ for the eight-digit number "62x2365y" to be divisible by 72, we need to check the divisibility rules for both 8 and 9, as 72 = 8 × 9.

Step 1: Check divisibility by 8

A number is divisible by 8 if its last three digits form a number that is divisible by 8. The last three digits of "62x2365y" are "65y".

We need to find a value of $y$ such that "65y" is divisible by 8.

• For $y = 0$, "650" is not divisible by 8.
• For $y = 1$, "651" is not divisible by 8.
• For $y = 2$, "652" is divisible by 8.
• For $y = 3$, "653" is not divisible by 8.
• For $y = 4$, "654" is not divisible by 8.
• For $y = 5$, "655" is not divisible by 8.
• For $y = 6$, "656" is divisible by 8.
• For $y = 7$, "657" is not divisible by 8.
• For $y = 8$, "658" is not divisible by 8.
• For $y = 9$, "659" is not divisible by 8.

So, $y$ can be either 2 or 6.

Step 2: Check divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits of "62x2365y" is $6 + 2 + x + 2 + 3 + 6 + 5 + y = 24 + x + y$.

We need $24 + x + y$ to be divisible by 9.

Case 1: $y = 2$

$24 + x + 2 = 26 + x$

For $26 + x$ to be divisible by 9, $x$ must be 7 (since $26 + 7 = 33$, which is divisible by 9).

Case 2: $y = 6$

$24 + x + 6 = 30 + x$

For $30 + x$ to be divisible by 9, $x$ must be 6 (since $30 + 6 = 36$, which is divisible by 9).

Step 3: Calculate $x^2 + y^2 - xy$

For $x = 7$ and $y = 2$:

$x^2 + y^2 - xy = 7^2 + 2^2 - 7 \cdot 2 = 49 + 4 - 14 = 39$

For $x = 6$ and $y = 6$:

$x^2 + y^2 - xy = 6^2 + 6^2 - 6 \cdot 6 = 36 + 36 - 36 = 36$

Among the given options, the correct value is 36.