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For any integer d > 0, let f(d) be the smallest possible integer that has exactly d positive divisors (so for example we have f(1)=1, f(5)=16, and f(6)=12). Prove that for every integer k \geq 0 the number f\left(2^k\right) divides f\left(2^{k+1}\right).

Proposed by Suhaimi Ramly, Malaysia

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