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Sum of fractional parts

Source: 2024 Israel National Olympiad (Gillis) P2

December 7, 2023
number theoryremaindersfractional partnational olympiad

Problem Statement

A positive integer xx satisfies the following: {x3}+{x5}+{x7}+{x11}=248165\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165} Find all possible values of {2x3}+{2x5}+{2x7}+{2x11}\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\} where {y}\{y\} denotes the fractional part of yy.
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