Question

patty tutors students in groups of 10. meanwhile, nellie 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
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Explanation:

Step1: Identify the problem type

We need to find the least common multiple (LCM) of 10 and 7 since we want the smallest number that is a multiple of both 10 and 7 (as both tutors have the same total number of students, which must be a multiple of their respective group sizes).

Step2: Prime factorize the numbers

• Prime factorization of 10: \(10 = 2\times5\)
• Prime factorization of 7: \(7 = 7\) (7 is a prime number)

Step3: Calculate the LCM

To find the LCM, we take the highest power of each prime number that appears in the factorizations.

• For prime 2: highest power is \(2^1\)
• For prime 5: highest power is \(5^1\)
• For prime 7: highest power is \(7^1\)

Then, LCM \(= 2\times5\times7 = 70\)

Answer:

70