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.
- (a) I only
- (b) II only
- (c) Both I and II ✓ UPSC's answer
- (d) Neither I nor II
Why the answer is (c)
• For any two numbers a and b, HCF × LCM = a × b.
• If a × b = a³, then b = a²: one number is the square of the other. II is always true.
• The difference is a² − a = a(a − 1), a product of two consecutive integers, which is always even. I is always true.
• Hence both: (c).
Why the other options are wrong
- (a) I only
- II is also true: the other number equals the square of the first.
- (b) II only
- I is also true: a(a − 1) is always even.
- (d) Neither I nor II
- Both statements follow from HCF × LCM = product of the numbers.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.