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).