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Number Theory : primes

Source: Iran 3rd round 2017 Number theory first exam-P1

August 9, 2017
number theoryDivisibilityprime numbersprimeIranIranMO

Problem Statement

Let nn be a positive integer. Consider prime numbers p1,…,pkp_1,\dots ,p_k. Let a1,…,ama_1,\dots,a_m be all positive integers less than nn such that are not divisible by pip_i for all 1≤i≤n1 \le i \le n. Prove that if m≥2m\ge 2 then 1a1+⋯+1am\frac{1}{a_1}+\dots+\frac{1}{a_m} is not an integer.
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