What is $|\{n\in N:n\mid p^2q\}|$?

Question

$$|\{n\in N:n\mid p^2q\}|$$ $p$ is an odd prime number, $q$ is a prime number and $p$ does not equal $q$.

I dont understand what they're asking me to find, is it to find a value of $n$ that divides into $p^2q$?

I need all the help i can get here, very new to this
thanks

No; they are asking you to compute the number of dicisors of $p^2q$. — José Carlos Santos, Jan 29 '18 at 15:15
i mean how would i work out the divisors in the first place? — , Jan 29 '18 at 15:17
@Chloe $p,q$ are prime numbers, so let's do an example: how many different numbers divide $3^2\cdot 2$? What about $5^2\cdot 7$? — cansomeonehelpmeout, Jan 29 '18 at 15:22

score 1 · Answer 1 · answered Jan 29 '18 at 15:17

1

What they ask for is in set-builder notation: the number of natural numbers $n$ such that $n$ divides $p^2q$, or the number of divisors of $p^2q$, or just $\tau(p^2q)$.

$p$ does not have to be an odd prime for us to figure this out, just that $p\ne q$. There are six divisors: $$1,p,p^2,q,pq,p^2q$$ So the required answer is 6.

answered Jan 29 '18 at 15:17

Parcly Taxel

103,344

the question says the p is an odd prime, or does is it not matter? – Jan 29 '18 at 15:18
@Chloe This turns out to be irrelevant (at least to the problem at hand). – Parcly Taxel Jan 29 '18 at 15:18
ok thanks alot! – Jan 29 '18 at 15:19

score 1 · Answer 2 · answered Jan 29 '18 at 15:25

The inner part of that expression inside the curly braces means the set of all positive integers which divide $p^2q$.

The outer part with "|...|" means to count the number of elements in that set.

The whole expression is asking you to count the number of divisors of $p^2q$.

What is $|\{n\in N:n\mid p^2q\}|$?

2 Answers2