How many positive divisors of 2000? I thought that the number of divisors of a number was the product of the indices in its factorisation, plus $2$ (for 1 and the number itself). For instance, $2000=2^{4} \cdot 5^{3}$, so it would have $4 \cdot 3 + 2 = 14$ divisors. 
Apparently, however, $2000$ actually has $20$ divisors. What am I doing wrong?

The factors are of the form of $2^x\cdot 5^y$ where $x$ takes value from $0$ to $4$ and $y$ takes values from $0$ to $3$. 
Hence there is a total of $(4+1)(3+1)=20$ factors.
Your method miss out number such as $5^y, y>0; 4^x, x>0$ and double counted $2000$.
