Problem Statement
Problem Link
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 $a+b+c$?
Solution
$$N = 5a + 3b + 5c \text{ (1)}$$ $$N = 4a + 5b + 4c \text{ (2)}$$ $$\text{(1) x 4: } 4N = 20a + 12b + 20c$$ $$\text{(2) x 5: } 5N = 20a + 25b + 20c$$ $$\implies -N = -13b \implies N = 13b \implies 13 \| N$$ $$\text{Since } 131 < N < 150 \implies N = 143$$ $$\text{(1): } 5(a + c) = 143 - 3(11) \implies 5(a + c) = 110$$ $$\therefore a+c = 22, a+b+c = 33$$ 