Asked by GURL IDK ANY THING
Jill is packing equal quantities of pretzels and crackers for snacks. Jill bags the pretzels
in groups of 6 and the cracks in groups of 9. What is the smallest number of crackers that she can pack?
in groups of 6 and the cracks in groups of 9. What is the smallest number of crackers that she can pack?
Answers
Answered by
GPT-4o mini
To find the smallest number of crackers that Jill can pack, we need to determine the least common multiple (LCM) of the two group sizes: 6 (for pretzels) and 9 (for crackers).
1. **Find the prime factorization of each number:**
- The prime factorization of \(6\) is \(2 \times 3\).
- The prime factorization of \(9\) is \(3^2\).
2. **Identify the highest power of each prime factor present in either number:**
- The prime \(2\) appears in \(6\) as \(2^1\).
- The prime \(3\) appears in \(6\) as \(3^1\) and in \(9\) as \(3^2\).
3. **Combine the highest powers:**
- For \(2\), the highest power is \(2^1\).
- For \(3\), the highest power is \(3^2\).
4. **Calculate the LCM:**
\[
\text{LCM} = 2^1 \times 3^2 = 2 \times 9 = 18
\]
The LCM of 6 and 9 is 18, which means the smallest number of crackers that Jill can pack (while also packing an equal quantity of pretzels) is \(\boxed{18}\).
1. **Find the prime factorization of each number:**
- The prime factorization of \(6\) is \(2 \times 3\).
- The prime factorization of \(9\) is \(3^2\).
2. **Identify the highest power of each prime factor present in either number:**
- The prime \(2\) appears in \(6\) as \(2^1\).
- The prime \(3\) appears in \(6\) as \(3^1\) and in \(9\) as \(3^2\).
3. **Combine the highest powers:**
- For \(2\), the highest power is \(2^1\).
- For \(3\), the highest power is \(3^2\).
4. **Calculate the LCM:**
\[
\text{LCM} = 2^1 \times 3^2 = 2 \times 9 = 18
\]
The LCM of 6 and 9 is 18, which means the smallest number of crackers that Jill can pack (while also packing an equal quantity of pretzels) is \(\boxed{18}\).