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Ukrainian National Mathematical Olympiad, 3rd Round

Algebra Difficulty 5.2 AIME, harder Prove it Ukraine

Olesya wrote down a natural number $N$ . After this, Andriy wrote down one sixth, one fifth, one fourth, one third and a half of $N$ . It turns out that the sum of all numbers that are written is integer. What is the least possible number that Olesya could write?

Solution

The sum is given by: $\left(\frac{1}{6} + \frac{1}{5} + \frac{1}{4} + \frac{1}{3} + \frac{1}{2}\right)N = \frac{29N}{20}$ , and therefore $N$ is divisible by $20$ and the result follows.

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