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

Q: Why do I need the LCM instead of just any common multiple?

The LCM gives you the smallest possible class size that works for all group divisions. Larger multiples like 72 or 108 would also work, but they're outside the given range of 29-39 students.

Q: Do I need to include groups of 2 in my LCM calculation?

Actually, no! Since any number divisible by 4 is automatically divisible by 2, you only need to find LCM(4, 6, 9). The pairs requirement is already satisfied.

Q: What if there were multiple numbers in the range that worked?

Then you'd need additional information to determine the exact answer. In this problem, 36 is the only multiple of 36 between 29 and 39, making it the unique solution.

Q: What does 'exact division' mean in this context?

It means no remainder when dividing students into groups. If there are 36 students and you make groups of 4, you get exactly 9 complete groups with zero students left over.

Least Common Multiple with Range Constraints

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?

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Step-by-step written solution

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Understand the problem

Step-by-step solution

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 $36$ .

Final Answer

$36$

Key Points to Remember

Essential concepts to master this topic

Divisibility Rule: Number must divide evenly by 4, 2, 6, and 9
LCM Method: Prime factorize: 4= $2^2$ , 6= $2×3$ , 9= $3^2$ , so LCM= $2^2×3^2=36$
Range Check: Verify 36 is between 29-39 and divides evenly: 36÷4=9, 36÷6=6, 36÷9=4 ✓

Common Mistakes

Avoid these frequent errors

Adding instead of finding LCM of group sizes
Don't add the group sizes 4+2+6+9=21! This gives a number that won't divide evenly by all group sizes. Always find the LCM by using prime factorization to get the smallest number divisible by all given numbers.

Practice Quiz

Test your knowledge with interactive questions

Will a number divisible by 6 necessarily be divisible by 3?

FAQ

Everything you need to know about this question

Why do I need the LCM instead of just any common multiple?

The LCM gives you the smallest possible class size that works for all group divisions. Larger multiples like 72 or 108 would also work, but they're outside the given range of 29-39 students.

Do I need to include groups of 2 in my LCM calculation?

Actually, no! Since any number divisible by 4 is automatically divisible by 2, you only need to find LCM(4, 6, 9). The pairs requirement is already satisfied.

What if there were multiple numbers in the range that worked?

Then you'd need additional information to determine the exact answer. In this problem, 36 is the only multiple of 36 between 29 and 39, making it the unique solution.

How do I find prime factorization quickly?

Start with the smallest prime (2) and keep dividing:

4 ÷ 2 = 2, then 2 ÷ 2 = 1, so 4 = $2^2$
6 ÷ 2 = 3, then 3 ÷ 3 = 1, so 6 = $2×3$
9 ÷ 3 = 3, then 3 ÷ 3 = 1, so 9 = $3^2$

What does 'exact division' mean in this context?

It means no remainder when dividing students into groups. If there are 36 students and you make groups of 4, you get exactly 9 complete groups with zero students left over.

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