Find the Class Size: Dividing Students into Groups of 8, 4, and 2

Q: Why do I need to check all three group sizes?

The problem states that all divisions must be exact with no students left out. Your answer must work for groups of 8, groups of 4, AND pairs (groups of 2).

Q: What if multiple numbers in the range work?

That's possible! You'd need to check each candidate by dividing it by 8, 4, and 2. Only numbers that divide evenly by all three are valid solutions.

Q: Do I need to find the LCM of 8, 4, and 2?

Yes! The LCM (Least Common Multiple) gives you the smallest number that works. Since $ \text{LCM}(8,4,2) = 8 $, you need multiples of 8.

Q: How do I verify my answer works for all group sizes?

$ 32 ÷ 8 = 4 $ groups (no remainder)$ 32 ÷ 4 = 8 $ groups (no remainder)$ 32 ÷ 2 = 16 $ pairs (no remainder)All divisions are exact, so 32 is correct!

Common Multiples with Range Constraints

A teacher divides the students in his class into group discussions.

On the first day he divides them into groups of 8.

On the second day he divides them into pairs.

On the third day, he divides them into groups of 4.

The division is exact and no students are left without a group.

Calculate the number of students in the class, given that the number of students in the class ranges from 29 to 39.

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Step-by-step written solution

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Understand the problem

Step-by-step solution

To determine the number of students, let's find all multiples of 8 within the given range of 29 to 39.

Calculating multiples of 8: $8 \times 1 = 8$ , $8 \times 2 = 16$ , $8 \times 3 = 24$ , $8 \times 4 = 32$ , $8 \times 5 = 40$ .

The relevant multiple of 8 that falls within the 29 to 39 range is $32$ .

Thus, the number of students in the class is $32$ .

This means that on any given day, all students can indeed be grouped as specified without any remainder.

Therefore, the solution to the problem is $32$ .

Final Answer

$32$

Key Points to Remember

Essential concepts to master this topic

Divisibility Rule: A number must divide evenly into all group sizes
Technique: Find LCM of 8, 4, and 2: LCM(8,4,2) = 8
Check: Verify 32 ÷ 8 = 4, 32 ÷ 4 = 8, 32 ÷ 2 = 16 groups ✓

Common Mistakes

Avoid these frequent errors

Only checking multiples of one group size
Don't just find multiples of 8 in the range = incomplete solution! This ignores that the number must also divide evenly by 4 and 2. Always verify the number works for ALL group sizes mentioned.

Practice Quiz

Test your knowledge with interactive questions

Determine if the following number is divisible by 3:

$$ 352 $$

FAQ

Everything you need to know about this question

Why do I need to check all three group sizes?

The problem states that all divisions must be exact with no students left out. Your answer must work for groups of 8, groups of 4, AND pairs (groups of 2).

What if multiple numbers in the range work?

That's possible! You'd need to check each candidate by dividing it by 8, 4, and 2. Only numbers that divide evenly by all three are valid solutions.

Do I need to find the LCM of 8, 4, and 2?

Yes! The LCM (Least Common Multiple) gives you the smallest number that works. Since $\text{LCM}(8,4,2) = 8$ , you need multiples of 8.

Why is 32 the only answer in the range?

Within 29-39, only 32 is a multiple of 8. Since 8 is already the LCM of all group sizes, 32 automatically works for groups of 4 and 2 as well.

How do I verify my answer works for all group sizes?

$32 ÷ 8 = 4$ groups (no remainder)
$32 ÷ 4 = 8$ groups (no remainder)
$32 ÷ 2 = 16$ pairs (no remainder)

All divisions are exact, so 32 is correct!

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