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2019 JBMO TST- North Macedonia, problem 5

Source: 2019 JBMO TST- North Macedonia

May 26, 2019

JMMO2019JuniorMacedonianumber theoryprime numbers

Problem Statement

Let

p_{1}

p_{2}

, ...,

p_{k}

be different prime numbers. Determine the number of positive integers of the form

p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}

\alpha_{i}

\in

\mathbb{N}

for which

\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}