Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?
$\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55$
As all five options are divisible by $5$, we may not use any pennies. (This is because a penny is the only coin that is not divisible by $5$, and if we used between $1$ and $4$ pennies, the sum would not be divisible by $5$.)
Hence the smallest coin we can use is a nickel, and thus the smallest amount we can get is $4\cdot 5 = 20$. Therefore the option that is not reachable is $\boxed{15}$ $\Rightarrow$ $(A)$.
We can verify that we can indeed get the other ones:

$25 = 10+5+5+5$
$35 = 10+10+10+5$
$45 = 25+10+5+5$
$55 = 25+10+10+10$