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Number theory Difficulty 2.0 Junior Find the answer United Kingdom

The numbers $p$ , $q$ , $r$ and $s$ satisfy the following equations: $p+2q+3r+4s=k$ , $4p=3q=2r=s$ . What is the smallest value of $k$ for which $p$ , $q$ , $r$ and $s$ are all positive integers?

(A) $20$ (B) $24$ (C) $25$ (D) $77$ (E) $154$

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Source: UK Mathematics Trust, licensed © UK Mathematics Trust; past papers published free at ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.