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?

The correct option is 8.

To find the number of students who play both cricket and football, we need to determine how many identification numbers between 1 and 100 are divisible by both 4 and 6.

A student plays football if their identification number is divisible by 4.
A student plays cricket if their identification number is divisible by 6.

For a student to play both games, their identification number must be divisible by both 4 and 6. The smallest positive integer that is divisible by both 4 and 6 is their Least Common Multiple (LCM).

Let us find the LCM of 4 and 6:
The multiples of 4 are 4, 8, 12, 16, 20, 24, ...
The multiples of 6 are 6, 12, 18, 24, ...
The smallest common multiple is 12. Therefore,

$$\text{LCM}(4,6)=12$$

This means that any identification number divisible by both 4 and 6 must be a multiple of 12. Now, we need to find how many multiples of 12 exist in the range from 1 to 100.

We can find this by dividing the total number of students (100) by 12 and taking the integer part (floor value) of the quotient:

$$\text{Number of students}=\lfloor \frac{100}{12}\rfloor$$

Calculating the division:
100 divided by 12 is 8 with a remainder of 4 (since 12 * 8 = 96).

Therefore, the multiples of 12 between 1 and 100 are:
12, 24, 36, 48, 60, 72, 84, and 96.

Counting these values, we find there are exactly 8 students who play both cricket and football.