Factors of a number? I'm studying for the SAT on collegeboard.org, and I came across the problem: What is the least positive integer that has the same number of positive factors as 175?

The answer was 12, explained as:

The number 175 is equal to 5 x 7 x 5. It has a total of 6 positive factors: 1, 5, 7, 25, 35, and 175. The numbers with 6 positive factors are of the form$p_1^2*p_2$or$p^5$. The least number of the first form is 12. The least number of the second form is 32. Thus, the answer is 12, which has the  positive factors 1, 2, 3, 4, 6, and 12.

I've never learned anything about the number of factors for a number, so this line: The numbers with 6 positive factors are of the form$p_1^2*p_2$or$p^5$ doesn't make sense to me. Out of context, I understand that $p_1$ is 2, but I don't understand where those two formulas came from or what p is supposed to be the set of.
$p_1$, $p_2$, and $p$ in your quote stand for primes - arbitrary primes. That is, if $p$ is any prime whatsoever, then $p^5$ has exactly 6 factors; and if $n$ has exactly 6 factors, then either $n=p^5$ for some prime $p$, or $n=p_1^2p_2$ for some primes $p_1$ and $p_2$. Everything rests on a formula for the number of factors: if $$n=p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r}$$ where the $p_i$ are distinct primes and the $a_1$ are positive integers, then the number of factors of $n$ is $$(a_1+1)(a_2+1)\times\cdots\times(a_r+1)$$ 

You can find this explained in any good introductory number theory textbook.
This seems a little hard for the SATs.  I assume there were multiple choices?  This might have been one case where you might have tried working backwards, counting the factors of each answer starting with the least.