Question

Assertion (A): The number of ways of writing $10800$ as the product of two positive integers is $30$.
Reason (R): The $10800$ is divisible by exactly three prime numbers.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

Answer: (B)

Here, $N = 10800 = 2^4\cdot 3^3 \cdot 5^2 = 2^{\alpha_1}3^{\alpha_2}5^{\alpha_3}$
$\therefore $ Number of ways in which $N$ can be resolved as a product of two factors
$ = \frac{1}{2}(\alpha_1 + 1)(\alpha_2 + 1)(\alpha_3 + 1)$
$(\because N$ is not a perfect square)
$ = \frac{1}{2}( 4 + 1)(3 + 1)(2 + 1) = 30$
And $N = 10800 = 2^4 \cdot 3^3 \cdot 5^2$ is divisible by exactly three primes $2, 3\, \&\, 5$
$\therefore $ Assertion (A) & Reason (R) both are true but Reason (R) is not the correct explanation of Assertion (A).