Let: $\displaystyle \delta(n)=0$ if 5 doesn't divide n, and $\displaystyle \delta(n)=1$ if 5 does divide n

The coefficient of $\displaystyle x^{100}$ stays untouched if we change $\displaystyle (1+x^{5}+x^{10}+x^{15}+...+x^{100})$ for $\displaystyle 1+x^{5}+x^{10}+x^{15}+...$

We have $\displaystyle 1+x^{5}+x^{10}+x^{15}+...=\sum_{k=0}^{\infty}{\del ta(n)\cdot{x^n}}$

Thus: $\displaystyle (1+x+x^{2}+...)(1+x^{5}+x^{10}+x^{15}+...)=\sum_{k =0}^{\infty}{\sum_{k=0}^n{\delta(k)}}\cdot{x^n}$

But: $\displaystyle \sum_{k=0}^n{\delta(k)}$ is the number of multiples of 5 between 0 and n, thus we have $\displaystyle

\sum\limits_{k = 0}^n {\delta \left( k \right)} = \left\lfloor {\tfrac{n}

{5}} \right\rfloor + 1

$ where $\displaystyle

\left\lfloor x \right\rfloor

$ is the floor function

So the coefficient of $\displaystyle x^{100}$ turns out to be $\displaystyle

\sum\limits_{k = 0}^{100} {\delta \left( k \right)} = \left\lfloor {\tfrac{{100}}

{5}} \right\rfloor + 1 = 21

$