Competition problem — Number theory

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Number theory Difficulty 6.0 National olympiad Prove it

4. Let $(a, b)=1, c$ be an integer. Prove: In the Cartesian coordinate system, on the line with the equation $a x+b y=c$ , any segment of length $\geqslant\left(a^{2}+b^{2}\right)^{1 / 2}$ (including endpoints) must contain a point whose coordinates are integers.

Solution

4. Let the solution be $x=x_{0}+b t, y=y_{0}-a t$ . The distance between the integer points given by two consecutive solutions (i.e., corresponding to $t, t+1$ ) is equal to $\left(a^{2}+b^{2}\right)^{1 / 2}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.