Question

a group of friends get together to make friendship bracelets. A package of embroidery floss can be shared equally among 3,5,or 6 friends with no strands left over.
What is the least strands the packet can? Suppose the package is part I could also be sure equally between two friends. Does this change the answer to part a does this answer change your answer to part why? Why not

Answer 1

To find the least number of strands in the package of embroidery floss that can be shared equally among 3, 5, or 6 friends, we need to determine the least common multiple (LCM) of these numbers.

1. **Finding the LCM of 3, 5, and 6:**

- The prime factorization of the numbers:
- \(3 = 3^1\)
- \(5 = 5^1\)
- \(6 = 2^1 \times 3^1\)

To find the LCM, we take the highest power of each prime factor:
- For \(2\), the highest power is \(2^1\) (from 6).
- For \(3\), the highest power is \(3^1\) (from 3 and 6).
- For \(5\), the highest power is \(5^1\) (from 5).

So,
\[
\text{LCM} = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30
\]

Thus, the least number of strands the package can have is **30**.

2. **Considering the condition of sharing equally among 2 friends:**

Now, we must check if 30 can also be divided equally among 2 friends.

To check this, we simply divide 30 by 2:
\[
30 \div 2 = 15
\]
Since 15 is an integer, 30 can be shared equally among 2 friends.

Therefore, the same answer (30 strands) holds true whether or not we include sharing among 2 friends.

In conclusion:
- The least number of strands the packet can have is **30**.
- Adding the condition of equal sharing among 2 friends does not change the answer; 30 is still the least common multiple that accommodates all conditions.