​If HCF of 768 and x⁶y²is 32 × x × y for natural numbers x ≥ 2, y ≥ 2, then what is the value of (x + y)?​

Correct option is A

Given:

HCF of 768 and x⁶y²is 32xy for natural numbers x ≥ 2, y ≥ 2.

Concept Used:

1. Highest Common Factor (HCF)

2. Prime factorization

Solution:

Prime factorization of 768:

768 = 2⁸× 3

Prime factorization of 32xy:

32xy = 2⁵× x × y

Prime factorization of x⁶y²:

x⁶y²= x⁶× y²

For HCF to be 32xy, the minimum power of each prime factor in both numbers should be considered:

2⁵× x × y = 32xy

Here, x and y must be such that:

2⁵× x × y is a factor of both 768 and x⁶y².

Since 32 = 2⁵,

Therefore, x and y must be:

x = 2

y = 3

Sum of x and y:

=> x + y = 2 + 3 = 5

∴ The value of (x + y) is 5.