Question

how many positive divisors does 212100 have

Answer 1

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

Answer 2

yes, got 2x2x3x5x5x7x101

Answer 3

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.

Answer 4

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 \[(2+1)\cdot(1+1)\cdot(2+1)\cdot(1+1)\cdot(1+1)=3\cdot2\cdot3\cdot2\cdot2=2^3\cdot3^2\]

Answer 5

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

Answer 6

Did that make sense?