UPSC Prelims 2026 · CSAT
Question 59: Basic Numeracy
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?
- IThe difference of the numbers is an even number.
- IIOne of the numbers is a perfect square.
Select the answer using the code given below.
AnswerUPSC’s answer is (c).
For two numbers, HCF × LCM equals their product. If the product ab is the cube of one of them, say a³, then b = a². The numbers are distinct, so a is not 1. Difference: a² − a = a(a − 1), a product of consecutive integers, so it is even. One number, a², is a perfect square. Both statements hold.
- I. The difference a(a − 1) is a product of consecutive integers, so even.
- II. One number is a², a perfect square.
How you could have got this
- If you were sure statement I is correct, (b) and (d) go: each of them leaves I out.
What you needed to know: The difference a(a − 1) is a product of consecutive integers, so even.
- Then, if you were sure statement II is correct, (a) goes: it leaves II out.
What you needed to know: One number is a², a perfect square.
Only (c) is left. This one needs every statement.
This follows UPSC’s provisional key; the final key could change it.
Which options go is worked out from UPSC’s key and the options, not by a model.
Why each option is right or wrong
- (a) Wrong: II also holds, since a² is a perfect square.
- (b) Wrong: I also holds; a(a − 1) is even.
- (c) Right: b = a² makes both true.
- (d) Wrong: both statements hold.
Only the provisional key (27 May 2026) was found; the final key replaces it when found.
Answer: UPSC’s key. Explanation: written by UPSC Grade, not by UPSC.
Source: UPSC, Civil Services (Preliminary) Examination 2026, General Studies Paper II (CSAT), Series A. The paper on upsc.gov.in.