Bob is putting the students of the sports club in groups. He finds out that if he makes three students in a group, two students will be left over. If he makes five students in a group, also 2 students will be left over. The total number of students is between 60 & 70. How many students are there in the sports club?
Let the total number of students be x.
Given that if Bob makes groups of three students, two students will be left over, we can write this as:
x ≡ 2 (mod 3)
Also, given that if Bob makes groups of five students, two students will be left over, we can write this as:
x ≡ 2 (mod 5)
Combining the two equations, we get:
x ≡ 2 (mod 15)
Now, we know that the total number of students is between 60 and 70. To find the number of students, we need to find the number that satisfies the above congruence within this range.
The numbers between 60 and 70 that satisfy the congruence x ≡ 2 (mod 15) are 62 and 77.
Therefore, there are 62 students in the sports club.