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?

Correct Option

A student plays football if their identification number is divisible by 4. A student plays cricket if their identification number is divisible by 6. To play both cricket and football, a student's identification number must be divisible by both 4 and 6.

This implies the identification number must be a multiple of the Least Common Multiple (LCM) of 4 and 6.

• Prime factorization of 4 = 2²
• Prime factorization of 6 = 2 × 3
• LCM(4, 6) = 2² × 3 = 12

Therefore, students who play both sports have identification numbers that are multiples of 12. We need to find the count of such numbers between 1 and 100.

The multiples of 12 within this range are: 12, 24, 36, 48, 60, 72, 84, 96.

Counting these numbers, there are 8 students who play both cricket and football.

Incorrect Options

Options 1 (4), 3 (10), and 4 (12) are incorrect because the calculation for the number of students whose identification numbers are divisible by both 4 and 6 (i.e., by 12) yields 8, not these values. For instance, 4 represents only a subset of common multiples, 10 is the count of numbers divisible by 6, and 12 is the LCM itself, not the count of students.