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, 2011

Number theory Difficulty 3.0 AMC 10/12 Prove it Canada

Positive integers $(x, y, z)$ form a Trenti-triple if $3x = 5y = 2z$ .
(a) Determine the values of $y$ and $z$ in the Trenti-triple $(50, y, z)$ .
(b) Show that for every Trenti-triple $(x, y, z)$ , $y$ must be divisible by 6.
(c) Show that for every Trenti-triple $(x, y, z)$ , the product $xyz$ must be divisible by 900.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.