how many positive divisors does 212100 have

Question

kinggeorge · Answer

First, we have to factor 212100 can you do that yourself?

anonymous · Answer

yes, got 2x2x3x5x5x7x101

kinggeorge · Answer

Right, so to write it using exponents, we have$$212100=2^2\cdot3\cdot5^2\cdot7\cdot101$$To find the number of divisors, we have to pick certain subsets of these exponentsso that we don't repeat any numbers.

kinggeorge · Answer

The number of choices we have for each exponent, is exactly $e+1$ where $e$ is a given exponent. 

Our exponents are $\{2, 1, 2, 1, 1\}$ respectively. That means we have to multiply $$(2+1)\cdot(1+1)\cdot(2+1)\cdot(1+1)\cdot(1+1)=3\cdot2\cdot3\cdot2\cdot2=2^3\cdot3^2$$

kinggeorge · Answer

Do the computations, you get that 212100 has exactly 72 positive divisors.

kinggeorge · Answer

Did that make sense?