Question

'a' and 'b' are prime numbers. Which of the following is true?
I. a? has three positive integer factors.
II. ab has four positive integer factors.
III. a' has four positive integer factors.
1) I and II only
2) II and III only
3) All of these
4) None of these​

Answer 1

Given 'a' and 'b' are prime numbers

1. Since 'a' is a prime number, there are only four integers (i.e., -a, -1, 1 and a) that divides 'a' out of which only two (1 and a) are positive, implying the falsity of statement 1.

2. ab is divisible by a, b, 1 and ab itself, these four factors are the only positive integer factors of ab, as a and b are both primes. So, statement 2 is true.

3. From 1 we saw that there are only two positive integer factors (1 and a) of 'a'. This implies that statement 3 is false.

From the above validation of the statements, we infer that, option 4 is correct.

Answer 2

Answer:-

GIVEN

1. a² has three positive integers

2. ab has four positive integers

3. a³ has four positive integers

VERIFIED RIGHT ANSWER !!!!!

The factors of a² are 1,a and a². The factors of ab are 1,a,b and ab.The factor of a³;are 2,a,a² and a³

HENCE :- ( ALL OF THESE IS THE ANSWER OPTION NUMBER 3 )