40 students attended the brilliant summer camp at Stanford University. "k" of the students at the camp are working on math problems and the others are working on non-math problems. One of the camp mentors noticed that whenever all the students were partitioned into 2 or more groups of equal size, they could never have the same number of math students in every group. How many different values could "k" have?
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Two thin and identical steel rods move with equal speeds v=1 m/s in opposite directions and collide longitudinally.The collision is perfectly elastic and the rods swap their velocities. How long does the collision last in seconds if the length of the rods is l=30 cm. Assume that the speed of sound in the steel rods is c=5000 m/s?
To find the number of different values that "k" could have, let's analyze the problem step by step.
We are given that there are 40 students in total, and "k" of them are working on math problems while the others are working on non-math problems. We need to determine how many different values "k" could have.
Considering the situation where all students are partitioned into two or more groups of equal size, we can understand that the total number of students must be divisible by the number of groups. In this case, the total number of students is 40.
Let's consider the possible number of groups we can form. Since we are interested in finding all possible values of "k," we can start with two groups.
If we have two groups, each group should have an equal number of students. Since 40 is divisible by 2, we can split the students equally, resulting in 20 students in each group.
Now, the mentor noticed that they can never have the same number of math students in every group. This means that the math students need to be distributed unevenly across the groups. Since we have 40 students in total and we want them to be evenly divided into two groups, we cannot distribute the math students with an equal number in each group.
If we add one math student to one group, we need to add one non-math student to the other group to maintain equal group sizes. But this will create a situation where one group has a different number of math students than the other group.
We can keep adding math students to one group and non-math students to the other group, but we will never achieve the same number of math students in each group.
Now, let's consider the situation with three groups. If we want to split 40 students into three equal groups, each group should have approximately 40/3 = 13.33 students. However, we cannot have a fraction of a student, so we need to round this number either up or down.
If we round it down, each group would have 13 students. If we distribute the math students evenly, each group would have (k/40)*13 math students. By checking all possible values of k from 1 to 39, we can see that it is never possible to have the same number of math students in every group.
If we round it up, each group would have 14 students. Again, if we distribute the math students evenly, each group would have (k/40)*14 math students. By checking all possible values of k from 1 to 39, we can see that it is never possible to have the same number of math students in every group.
Based on the analysis, we can conclude that it is impossible to divide the 40 students into two or more groups of equal size with the same number of math students in each group.
Therefore, there are 0 different values for "k" that satisfy this condition.