To determine the least number of students the class can have such that they can be divided into groups of 3, 4, 6, and 9, we need to find the Least Common Multiple (LCM) of these numbers. The LCM of a set of integers is the smallest positive integer that is divisible by each of the integers.
Let's find the LCM step-by-step:
1. List the prime factors of each number:
- 3: [tex]\(3\)[/tex]
- 4: [tex]\(2^2\)[/tex]
- 6: [tex]\(2 \times 3\)[/tex]
- 9: [tex]\(3^2\)[/tex]
2. Identify the highest power of each prime present in the factorizations:
- The highest power of 2: [tex]\(2^2\)[/tex] (from 4)
- The highest power of 3: [tex]\(3^2\)[/tex] (from 9)
3. Combine these highest powers to get the LCM:
- [tex]\(2^2 = 4\)[/tex]
- [tex]\(3^2 = 9\)[/tex]
Therefore, the LCM is:
[tex]\[
4 \times 9 = 36
\][/tex]
Hence, the least number of students the class can have, so that they can be divided into groups of 3, 4, 6, and 9, is 36 students.
