Total number of factors of a number Can you please explain how can we derive the total number of factors of a composite number using the concept of combinations ? Thanks in advance !
 A: A composite number $n$ can be written as $n=p_1^{a_1}\cdots p_k^{a_k}$. The number of factors of $n$ is the product of the number of factors of $p_i^{a_i}$ for each $i=1,\ldots,k$. (This is because it is a multiplicative function) It is easy to count the factors of $p^a$: they are simply the powers $p^0,p^1,\ldots,p^a$. Thus, there are $a+1$ such factors.
Applying this for each value of $i$, we have that the number of factors of $n$ is $\prod\limits_{i=1}^k(a_i+1)$.

To see this using a more combinatorial viewpoint, consider that a factor of $n$ is a number of the form $d=p_1^{b_1}\cdots p_k^{b_k}$, where each $b_i$ satisfies $0\le b_i\le a_i$. Thus, we have $a_i+1$ choices for each $b_i$, and the result follows.
A: By the fundamental theorem of arithmetic, every natural is the product of primes with a certain multiplicity.
Then every factor of the initial number is a product of the same primes with a (possibly) lower multiplicity.
Hence you need to form all combinations of the possible multiplicities.

$$360=2^33^25^1\to000,100,200,300,010,110,210,310,020,120,220,320,001,101,201,301,011,111,211,311,021,121,221,321$$
$(3+1)\cdot(2+1)\cdot(1+1)$ of them.
