UPSC Prelims 2026 CSAT, question 59

Topic: Number system

Q59

If the product of the HCF and LCM of two distinct numbers is the cube of one of the numbers, then which of the following statements is/are correct?

I. The difference of the numbers is an even number.

II. One of the numbers is a perfect square.

Select the answer using the code given below.

Solution.

Answer: (c) Both I and II

For any two positive integers $a$ and $b$ , $\operatorname{HCF}(a, b) \times \operatorname{LCM}(a, b) = a \times b$ ; since $a \times b = a^3$ with $a \ne b$ , we have $b = a^2$ (with integer $a \ge 2$ ).
Statement I is correct: the difference of the two numbers is $a^2 - a = a(a - 1)$ , which is the product of two consecutive integers and is therefore always even.
Statement II is correct: since $b = a^2$ , one of the two numbers is always a perfect square.