Civil Services Prep

Prelims 2026 · CSAT / Quantitative Aptitude · Question 59

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?<br/><br/>I. The difference of the numbers is an even number.<br/>II. One of the numbers is a perfect square.<br/><br/>Select the answer using the code given below.<br/>

  1. I only
  2. II only
  3. Both I and II
  4. Neither I nor II

Answer

Both I and II

For any two numbers, HCF × LCM = product of the numbers. If this equals the cube of one number, say a³, then ab = a³, so b = a². Thus the numbers are a and a². Their difference is a² - a = a(a-1), which is always even. Also, one number is clearly a perfect square, namely a². So both statements are correct.

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