यदि दो भिन्न संख्याओं के HCF और LCM का गुणनफल, उनमें से एक संख्या का घन है, तो निम्नलिखित में से कौन-सा/कौन-से कथन सही है/हैं?
- I.संख्याओं के बीच का अंतर एक सम संख्या है।
- II.संख्याओं में से एक संख्या एक पूर्ण वर्ग है।
नीचे दिए गए कूट का प्रयोग कर उत्तर चुनिए।
Read this question in English
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 · (d) Neither I nor II
उत्तर और व्याख्या · explanation in English
Answer: (c) I और II दोनों
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). ·