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 student plays cricket as well as football if their identification number is divisible by both 4 and 6. For a number to be divisible by both 4 and 6, it must be divisible by their Least Common Multiple (LCM). LCM(4, 6): Prime factorization of 4 = 2^2 Prime factorization of 6 = 2 x 3 LCM(4, 6) = 2^2 x 3 = 4 x 3 = 12. So, we need to find how many numbers between 1 and 100 are divisible by 12. These numbers are: 12, 24, 36, 48, 60, 72, 84, 96. To count them, we can divide 100 by 12: 100 / 12 = 8 with a remainder of 4. The integer part of the result, 8, gives the count of multiples of 12 up to 100. Therefore, there are 8 students who play both cricket and football. Analyzing the options: A) 4: Incorrect. This would be the case if it were divisible by 24 (100/24 = 4). B) 8: Correct. There are 8 numbers between 1 and 100 that are multiples of 12. C) 10: Incorrect. This would be the case if it were divisible by 10 (100/10 = 10). D) 12: Incorrect. This is the LCM, not the count. The final answer is B.