The natural number 2 is :

Question

Assertion (A) : \((a + \sqrt{b})(a - \sqrt{b})\) is a rational number, where a and b are positive integers.
Reason (R) : Product of two irrationals is always rational.

Answer 1

This question asks us to place the number 2 into one of four boxes: prime, composite, both, or neither. Instead of quoting the definitions and applying them once, let's test the number 2 directly against each option and rule three of them out.

a prime number: List every positive number that divides 2 exactly. Only 1 and 2 do the job, nothing else works. A number with exactly two divisors, 1 and itself, is by definition prime. Since 2 has exactly these two divisors and no more, this box fits.
a composite number: A composite number needs at least one divisor besides 1 and itself, for example 4 has the extra divisor 2, or 9 has the extra divisor 3. We already listed the divisors of 2 above, there are only two of them, so there is no room for a third divisor. This box does not fit.
prime as well as composite: For this to be true, 2 would have to satisfy both definitions at once, which we just showed is impossible since it fails the composite test. So a number cannot be both, and this box is wrong for a stronger reason too: no number is ever both prime and composite by definition, the two categories are built to be separate.
neither prime nor composite: This description belongs only to the number 1, which has just a single divisor (itself). Since 2 already qualifies as prime, it cannot also belong in this leftover category.

Only the first box survives the test: 2 is a prime number, and in fact it is the only even prime number that exists, because every other even number is divisible by 2 and by itself, giving it a third divisor and making it composite.

Let's summarize:

A number is prime when it has exactly two divisors, 1 and itself.
2 has exactly those two divisors and nothing more, so it is prime, not composite, and not "neither".

So the natural number 2 is a prime number, which is option (A).

Answer 2

This question asks us to place the number 2 into one of four boxes: prime, composite, both, or neither. Instead of quoting the definitions and applying them once, let's test the number 2 directly against each option and rule three of them out.

a prime number: List every positive number that divides 2 exactly. Only 1 and 2 do the job, nothing else works. A number with exactly two divisors, 1 and itself, is by definition prime. Since 2 has exactly these two divisors and no more, this box fits.
a composite number: A composite number needs at least one divisor besides 1 and itself, for example 4 has the extra divisor 2, or 9 has the extra divisor 3. We already listed the divisors of 2 above, there are only two of them, so there is no room for a third divisor. This box does not fit.
prime as well as composite: For this to be true, 2 would have to satisfy both definitions at once, which we just showed is impossible since it fails the composite test. So a number cannot be both, and this box is wrong for a stronger reason too: no number is ever both prime and composite by definition, the two categories are built to be separate.
neither prime nor composite: This description belongs only to the number 1, which has just a single divisor (itself). Since 2 already qualifies as prime, it cannot also belong in this leftover category.

Only the first box survives the test: 2 is a prime number, and in fact it is the only even prime number that exists, because every other even number is divisible by 2 and by itself, giving it a third divisor and making it composite.

Let's summarize:

A number is prime when it has exactly two divisors, 1 and itself.
2 has exactly those two divisors and nothing more, so it is prime, not composite, and not "neither".

So the natural number 2 is a prime number, which is option (A).

The natural number 2 is :

Show Hint

The Correct Option is A

Solution and Explanation

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