1 100 \frac{1}{100}

Number Theory Level 3

Without calculating decide that the statement below is true or false!

  • If 1 1 + 1 2 + 1 3 + ⋯ + 1 100 = m n \dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\dots+\dfrac{1}{100}=\dfrac{m}{n} , where m m and n n are coprime positive integers, then m m is divisible by 505 505 .
False True

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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1 solution

This is not a solution, but proving the statements below, you get the solution.

1. If p p is a prime, then p ∣ m p|m , where m m and n n are coprime, and 1 1 + 1 2 + 1 3 + ⋯ + 1 p − 1 = m n \dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\dots+\dfrac{1}{p-1}=\dfrac{m}{n}

2. 5 ∣ m 5|m , where m m and n n are coprime, and 1 1 + 1 2 + 1 3 + ⋯ + 1 100 = m n \dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\dots+\dfrac{1}{100}=\dfrac{m}{n}

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