Question

If the square root of p2 is an integer greater than 1, which of the following must be true? I. p2 has an odd number of positive factors II. p2 can be expressed as the product of an even number of positive prime factors III. p has an even number of positive factors?

Answer 1

Option | and Option || is True

Explanation:

Given:

If the square root of
`p^(2)` is an integer greater than 1,

Lets p = 2, 3, 4, 5, 6, 7..........

Solution:

Now we check all option for
`p^(2)`

Option |.

`p^(2)` has an odd number of positive factors.

Let
`p=2`

The positive factor of
`2^(2)=4=1,2,4`

Number of factor is 3

Let
`p=3`

The positive factor of
`3^(2)=9=1,3,9`

Number of factor is 3

So,
`p^(2)` has an odd number of positive factors.

Therefore, 1st option is true.

Option ||.

`p^(2)` can be expressed as the product of an even number of positive prime factors

Let
`p=2`

The positive factor of
`2^(2)=4=1,2,4`

`4=2* 2`

Let
`p=3`

The positive factor of
`3^(2)=9=1,3,9`

`9=3* 3`

So, it is expressed as the product of an even number of positive prime factors,

Therefore, 2nd option is true.

Option |||.

p has an even number of positive factors

Let
`p=2`

Positive factor of
`2=1,2`

Number of factor is 2.

Let
`p=4`

Positive factor of
`4=1,2,4`

Number of factor is 3 that is odd

So, p has also odd number of positive factor.

Therefore, it is false.

Therefore, Option | and Option || is True.

Option ||| is false.