Let S be a set of first 10 natural numbers. What is the possible number of pairs (a, b) where a, b ∈ S and a ≠ b such that the product ab (>12) leaves remainder 4 when divided by 12?

This question was previously asked in
CDS Maths Previous Paper 2 (Held On: 23 Oct 2016)
View all CDS Papers >
  1. 4
  2. 6
  3. 8
  4. 10

Answer (Detailed Solution Below)

Option 3 : 8
Free
UPSC CDS 01/2025 General Knowledge Full Mock Test
7.9 K Users
120 Questions 100 Marks 120 Mins

Detailed Solution

Download Solution PDF

S is a set of first 10 natural numbers;

∴ S = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)

∵ a, b ∈ S and a ≠ b such that ab > 12;

∴ The product ab can’t be any 3 digit number (it will be 2 digit number)

∴ Product ab (>12) that leaves remainder 4 when divided by 12 = (12n + 4);

∴ All such number (2 digit only) will be 16, 28, 40, 52, 64, 76, 88

Now classifying these numbers as the product ab in the pair (a, b) where a, b ∈ S a ≠ b:

16: (2, 8), (8, 2)

28: (4, 7), (7, 4)

40: (5, 8), (8, 5), (4, 10), (10, 4)

52: None

64: None

76: None

88: None

∴ Number of total pairs = 8

Latest CDS Updates

Last updated on May 29, 2025

-> The UPSC CDS 2 Notification has been released at upsconline.gov.in. for 453 vacancies.

-> Candidates can apply online from 28th May to 17th June 2025.

-> The CDS 2 Exam will be held on 14th September 2025.

-> Attempt UPSC CDS Free Mock Test to boost your score.

-> The selection process includes Written Examination, SSB Interview, Document Verification, and Medical Examination.  

-> Refer to the CDS Previous Year Papers to enhance your preparation. 

Get Free Access Now
Hot Links: teen patti noble teen patti real cash game teen patti online game teen patti master apk