Jill is packing equal quantities of pretzels and crackers for snacks. Jill bags the pretzels

Question

in groups of 6 and the cracks in groups of 9. What is the smallest number of crackers that she can pack?

Answer 1

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 $\boxed{{\displaystyle 18}}$.