Let $c$ and $k$ be positive integers such that $c^2 + ck + c = 40$ and $3k^2 + ck + k = 100$. What is the value of $\dfrac{c + 2k}{2}$?
Answer and solution
Correct answer: D
Solution
First equation: $c(c + k + 1) = 40$. Try values of $c$:
$c = 1 \Rightarrow k = 38$ ✗; $c = 2 \Rightarrow k = 17$ ✗; $c = 3 \Rightarrow k = \tfrac{28}{3}$ ✗
$c = 4 \Rightarrow 4k = 20 \Rightarrow k = 5$. Check: $3(25) + 20 + 5 = 100$ ✓
$\dfrac{c + 2k}{2} = \dfrac{4 + 10}{2} = 7$
$c = 1 \Rightarrow k = 38$ ✗; $c = 2 \Rightarrow k = 17$ ✗; $c = 3 \Rightarrow k = \tfrac{28}{3}$ ✗
$c = 4 \Rightarrow 4k = 20 \Rightarrow k = 5$. Check: $3(25) + 20 + 5 = 100$ ✓
$\dfrac{c + 2k}{2} = \dfrac{4 + 10}{2} = 7$