Question

Emilia is creating gift baskets with scented soaps and bottles of bubble bath. She has 16 scented soaps and 12 bottles of bubble bath. If she wants all the gift baskets identical without any items left over, what is the greatest number of gift baskets Emilia can make?

Answer 1

To determine the greatest number of identical gift baskets Emilia can make without any items left over, we need to find the greatest common divisor (GCD) of the two quantities of items she has: 16 scented soaps and 12 bottles of bubble bath.

First, we will find the GCD of 16 and 12 by finding their prime factorizations.

- The prime factorization of 16 is:
\[
16 = 2^4
\]
- The prime factorization of 12 is:
\[
12 = 2^2 \times 3^1
\]

Next, we identify the common prime factors and their lowest powers from each factorization.

- The prime factor \(2\) appears in both:
- In 16, the power is \(4\) (\(2^4\)).
- In 12, the power is \(2\) (\(2^2\)).

For \(2\), we take the smaller power, which is \(2^2\).

- The prime factor \(3\) appears in 12 but not in 16, so we do not include it in the GCD.

Now we calculate the GCD:
\[
GCD = 2^2 = 4
\]

This means the greatest number of identical gift baskets Emilia can make, using all the items without leftovers, is \(4\).

To verify, we can check how many soaps and bottles of bubble bath each basket would contain:
- For 4 baskets:
- Each basket would contain \( \frac{16}{4} = 4 \) scented soaps.
- Each basket would contain \( \frac{12}{4} = 3 \) bottles of bubble bath.

Since these values are whole numbers and there are no leftovers, our answer is confirmed.

Thus, the greatest number of gift baskets Emilia can make is:
\[
\boxed{4}
\]