Find the number of factors in the expression $12^5\cdot 13^{15}\cdot 5^{10}$

Question

After doing prime factorisation of 12:

There are ten 2s, five 3s, fifteen 13s and ten 5s in this number

A distinct factor of the number is created by taking 0 to 10 2s, 0 to 5 3s, 0 to 15 13s and 0 to 10 5s, and multiplying the choosen numbers together.

So there's 11 ways of picking 2s, 6 ways of picking 3s, 16 ways of picking 13s and 11 ways of picking 5s.

So number of factors = 11*6*16*11=11616

However, this answer is wrong as it is none of the options. What am I doing wrong here?

Answer 1

Turning a couple of comments into an answer, your solution to $12^5\cdot13^{15}\cdot5^{10}=2^{10}\cdot3^5\cdot13^{15}\cdot5^{10}$ is entirely correct. It's possible there was a typo in the problem, and that it meant to ask about the number of factors in $12^5\cdot13^5\cdot5^{10}=2^{10}\cdot3^5\cdot13^5\cdot5^{10}$, in which case the answer is $11\cdot6\cdot6\cdot11=4536$, which is one of the options.

Answer 2

The correct answer is not on their list.

As you have mentioned ,your number is $$(3^5)(2^{10})(13^{15})(5^{10})$$

Therefore you have $$(6)(11)(16)(11)=11616$$ divisors.