Let p and q be two natural numbers such that (p + q)^{p+q} is divisible by 512. What is the least value of (p+q)?

Explanation

We check the options for (p+q)^{p+q} being divisible by 512 = 2^9. If p+q = 4, 4^4 = 256 (not divisible). If p+q = 6, 6^6 = 2^6 \times 3^6 (not divisible). If p+q = 8, 8^8 = (2^3)^8 = 2^{24}, which is clearly divisible by 2^9. Hence, 8 is the least value.

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