If HCF of 768 and x^{6}y^{2} is 32xy for natural numbers x \geq 2, y \geq 2, then what is the value of (x+y)?

Explanation

768 = 2^8 \times 3. The HCF is 32xy = 2^5xy. For this to be the HCF with 768, x and y must be composed of the prime factors of 768 (which are 2 and 3). Since x \geq 2, y \geq 2, \{x, y\} = \{2, 3\}. Thus, x+y = 2+3 = 5.

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