Final answer:
To solve the problem, we need to set up a system of equations and find the values that satisfy it. Solving the system of equations, we find that the total number of sheets is 64. Dividing this by the number of sheets each student receives gives us the possible number of students: 64, 32, 16, 8, 4, 2, and 1.
Step-by-step explanation:
To determine the number of students, we can use the concept of divisibility and solve a system of equations. Let's assume the total number of sheets of colored paper is 'x'.
If the sheets are divided equally by 4, there are 10 sheets left. This means 'x' is 10 more than a multiple of 4, so we can write the equation: x = 4n + 10, where 'n' is an integer.
If the sheets are divided equally by 6, there is a shortage of 8 sheets. This means 'x' is 8 less than a multiple of 6, so we can write the equation: x = 6m - 8, where 'm' is an integer.
We can now solve this system of equations to find the value of 'x', which will represent the total number of sheets. Once we have 'x', we can then determine the number of students by dividing 'x' by the number of sheets each student receives.
Let's solve the system of equations:
4n + 10 = 6m - 8
4n - 6m = -18
Shifting the terms:
4n = 6m - 18
Dividing both sides by 2:
2n = 3m - 9
2n - 3m = -9
This equation represents the same relationship as the previous one, but with different coefficients. It's easier to solve this equation:
2n - 3m = -9
Let's make 'n' the subject of this equation:
2n = 3m - 9
Dividing both sides by 2:
n = 1.5m - 4.5
Since 'n' and 'm' are integers, let's find integer values for 'm' that satisfy this equation. By trying different values, we find that when 'm' equals 12, 'n' equals 15. Therefore, 'x' equals:
x = 6m - 8 = 6(12) - 8 = 72 - 8 = 64
Now, we can determine the number of students by dividing 'x' by the number of sheets each student receives. Assuming each student receives 's' sheets, we have:
Number of students = x / s = 64 / s
Since we want the number of students to be a whole number, 's' must be a factor of 64. The factors of 64 are 1, 2, 4, 8, 16, 32, and 64.
Therefore, the possible number of students are 64, 32, 16, 8, 4, 2, and 1.