16. The numbers $p, q, r$ and $s$ satisfy the following equations:
$$
p+2 q+3 r+4 s=k \quad 4 p=3 q=2 r=s .
$$

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
Solution
D

From the set of equations $4 p=3 q=2 r=s$, we have $q=\frac{4}{3} p, r=2 p$ and $s=4 p$.
We require $p$ and $q$ to be positive integers. Therefore, from the equation $q=\frac{4}{3} p$ we deduce that $p$ has to be an integer which is a multiple of 3 . We also note that if $p$ is a positive integer, then so also will be $r$ and $s$.

It also follows that
$$
\begin{aligned}
k & =p+2 q+3 r+4 s=p+2\left(\frac{4}{3} p\right)+3(2 p)+4(4 p) \\
& =p+\frac{8}{3} p+6 p+16 p=\frac{77}{3} p .
\end{aligned}
$$

Because $p$ is a positive integer which is a multiple of 3 , its smallest value is 3 . Therefore the smallest value of $k$ is $\frac{77}{3} \times 3$, which equals 77 .