Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels,dimes, and quarters, whose total value is $8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank?
$\text {(A) } 0 \qquad \text {(B) } 13 \qquad \text {(C) } 37 \qquad \text {(D) } 64 \qquad \text {(E) } 83$
Where $a,b,c$ is the number of nickels, dimes, and quarters, respectively, we can set up two equations:
\[(1)\ 5a+10b+25c=835\ \ \ \ (2)\ a+b+c=100\]
Eliminate $a$ by subtracting $(2)$ from $(1)/5$ to get $b+4c=67$. Of the integer solutions $(b,c)$ to this equation, the number of dimes $b$ is least in $(3,16)$ and greatest in $(67,0)$, yielding a difference of $67-3=\boxed{\textbf{(D)}\ 64}$.