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.
Answer & explanation
Answer: (c) Both I and II
HCF × LCM equals the product of the two numbers, so a × b = a³ means b = a². The numbers are a and a²: their difference a(a − 1) is a product of consecutive integers and hence even, and a² is a perfect square.
- For any two numbers, HCF × LCM = a × b.
- a × b = a³ gives b = a² (a ≠ 1, as the numbers are distinct).
- Difference = a² − a = a(a − 1), a product of two consecutive integers, so it is even.
- a² is a perfect square.
- Check: 3 and 9 — HCF 3, LCM 9, product 27 = 3³; difference 6, and 9 is a square.
- ✓ I a² − a = a(a − 1) is always even.
- ✓ II The second number is a², a perfect square.
Remember · HCF × LCM = product of the two numbers — the first thing to use in any HCF–LCM item.
Question and answer: UPSC's provisional GS Paper II (2026, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·