Question

If p and q are distinct primes. Find the number of positive divisors of
p^2 * q^2

Answer 1

The number of positive divisors are 7.

Explanation:

Since it is given that p and q are prime numbers thus they will not have any divisors other that itself and 1

The number
`p^(2)* q^(2)` can be represented as

`p^(2)* q^(2)=p* (p* q^(2))\\p^(2)* q^(2)=(p* q)* (p* q)\\p^(2)* q^(2)=(p^(2)* q)* q\\`

Thus the number
`(pq)^(2)` can be divided by:

1)
`1`

2)
`p`

3)
`q`

4)
`pq`

5)
`p* q^(2)`

6)
`p^(2)* q`

7)
`(pq)^(2)`

Thus the number of positive divisors are 7.

Answer 2

9

Explanation:

We know that for
`p^mq^n` where p and q are different prime numbers the number of positive divisors are (m+1) (n+!)

We have given
`p^2q^2`

So here m=2 and n=2

So number of positive divisors
`=(2+1)* (2+1)` =9

So the number of positive divisors of
`p^2q^2` is 9