In a school every student is assigned a unique identification number. A student is a football player if and only if the identification number is divisible by 4, whereas a student is a cricketer if and only if the identification number is divisible by 6. If every number from 1 to 100 is assigned to a student, then how many of them play cricket as well as football ?
- (a) 4
- (b) 8 ✓ UPSC's answer
- (c) 10
- (d) 12
Why the answer is (b)
• A student plays both cricket and football if their identification number is divisible by both 4 and 6.
• The least common multiple (LCM) of 4 and 6 is 12.
• Therefore, we need to count how many numbers from 1 to 100 are divisible by 12.
• The multiples of 12 in this range are 12, 24, 36, 48, 60, 72, 84, and 96.
• There are exactly 8 such numbers.
• Thus, 8 students play both cricket and football, which corresponds to option (b).
Why the other options are wrong
- (a) 4
- Option (a) is incorrect because there are 8 numbers divisible by 12, not 4.
- (c) 10
- Option (c) is incorrect because 10 is not the count of multiples of 12 between 1 and 100.
- (d) 12
- Option (d) is incorrect because 12 is the LCM itself, not the count of multiples within the range.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2019, held on 2 June 2019. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.