Question

Assertion :The number of ways of writing 1400 as the product of two positive integers is 12 Reason: The number $N=1400$ is not a perfect square.

• 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

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A),

Here $N=1400=2^3{5}^2{7}^1=2^{{\alpha }_1}{5}^{{\alpha }_2}{7}^{{\alpha }_3}$ is not a perfect square, so Reason (R) is true
Again the number of ways in which $N=1400$ can be resolved as a product of two factors is
given by $\frac{1}{2}({\alpha }_1+1)({\alpha }_2+1)({\alpha }_3+1)$ $=\frac{1}{2}(3+1)(2+1)(1+1)=12$, Since $N$ is not a perfect square.
Hence option 'A' is correct choice.
Note: If $N$ is perfect square then number of divisors is given by, $\frac{1}{2}\left[\right({\alpha }_1+1\left)\right({\alpha }_2+1\left)\right({\alpha }_3+1)+1]$