Question
Vijay tutors students in groups of 5. Meanwhile, Philip tutors students in groups of 7. While discussing their work, the tutors realize that they both work with the same total number of students. What is the smallest number of students each can have?
students
students
Answers
GPT-4o mini
To find the smallest number of students that both Vijay and Philip can have while keeping their group sizes in consideration, we need to find the least common multiple (LCM) of the two group sizes: 5 (for Vijay) and 7 (for Philip).
The LCM of two numbers is the smallest number that is a multiple of both. We can calculate the LCM of 5 and 7 by using the fact that they are both prime numbers.
The multiples of 5 are:
- 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ...
The multiples of 7 are:
- 7, 14, 21, 28, 35, 42, 49, 56, ...
The smallest number that appears in both lists is 35.
Thus, the least common multiple of 5 and 7 is:
\[
\text{LCM}(5, 7) = 35
\]
Now, since both Vijay and Philip work with the same total number of students, we conclude that the smallest number of students each can have is \( \boxed{35} \).
The LCM of two numbers is the smallest number that is a multiple of both. We can calculate the LCM of 5 and 7 by using the fact that they are both prime numbers.
The multiples of 5 are:
- 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ...
The multiples of 7 are:
- 7, 14, 21, 28, 35, 42, 49, 56, ...
The smallest number that appears in both lists is 35.
Thus, the least common multiple of 5 and 7 is:
\[
\text{LCM}(5, 7) = 35
\]
Now, since both Vijay and Philip work with the same total number of students, we conclude that the smallest number of students each can have is \( \boxed{35} \).