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Algebra Difficulty 3.0 AMC 10/12 Find the answer Canada

There are four unequal, positive integers $a$ , $b$ , $c$ , and $N$ such that $N = 5a + 3b + 5c$ . It is also true that $N = 4a + 5b + 4c$ and $N$ is between $131$ and $150$ . What is the value of $a+b+c$ ?

(A) $13$ (B) $17$ (C) $22$ (D) $33$ (E) $36$

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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.