Solve for Class Size: Divisibility by 4, 2, 6, and 9 in Student Groups

A teacher divides the students in their class into discussion groups.

On the first day, she divides them into groups of 4.

On the second day, she divides them into pairs.

On the third day, she divides them into groups of 6.

On the fourth day, she divides them into groups of 9.

The division is exact and no students are left without a group. The number of students in the class varies from 29 to 39.

How many students are in the class?

To solve this problem, identify the least common multiple (LCM) of the group sizes required: 4, 6, and 9.

• Calculate the LCM of 4, 6, and 9:
  • Prime factorize each number:
    • 4 = $2^2$
    • 6 = $2 \times 3$
    • 9 = $3^2$
  • The LCM is found by taking the highest power of each prime number present in any factorization: $(2^2 \times 3^2 = 36)$.
• Within the range 29 to 39, check which numbers are divisible by 36.
• The only number within this range that is exactly divisible by 36 is 36 itself.

Therefore, the number of students in the class is $36$.