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?
Answers
GPT-4o mini
To determine the greatest number of identical gift baskets Emilia can create without having any items left over, we need to find the greatest common divisor (GCD) of the two quantities she has: 16 scented soaps and 12 bottles of bubble bath.
We will start by finding the prime factorization of each quantity:
1. Factorizing 16:
\[
16 = 2^4
\]
2. Factorizing 12:
\[
12 = 2^2 \times 3^1
\]
Next, we find the GCD by taking the lowest power of each common prime factor:
- For the prime number \(2\):
- In 16, the power of 2 is 4.
- In 12, the power of 2 is 2.
- The minimum of these powers is \(2^2\).
- The prime number \(3\) does not appear in the factorization of 16. Therefore, it does not contribute to the GCD.
Now, we calculate the GCD:
\[
\text{GCD}(16, 12) = 2^2 = 4
\]
Thus, the greatest number of identical gift baskets that Emilia can make, with no items left over, is \( \boxed{4} \).
To check, we can verify how many soaps and bottles go into each basket when we divide the items accordingly:
- If she makes 4 baskets, then she would put:
- \( \frac{16 \text{ soaps}}{4 \text{ baskets}} = 4 \text{ soaps per basket} \)
- \( \frac{12 \text{ bottles}}{4 \text{ baskets}} = 3 \text{ bottles per basket} \)
This confirms that each of the 4 baskets can have 4 soaps and 3 bottles of bubble bath, utilizing all items without any remainder.
We will start by finding the prime factorization of each quantity:
1. Factorizing 16:
\[
16 = 2^4
\]
2. Factorizing 12:
\[
12 = 2^2 \times 3^1
\]
Next, we find the GCD by taking the lowest power of each common prime factor:
- For the prime number \(2\):
- In 16, the power of 2 is 4.
- In 12, the power of 2 is 2.
- The minimum of these powers is \(2^2\).
- The prime number \(3\) does not appear in the factorization of 16. Therefore, it does not contribute to the GCD.
Now, we calculate the GCD:
\[
\text{GCD}(16, 12) = 2^2 = 4
\]
Thus, the greatest number of identical gift baskets that Emilia can make, with no items left over, is \( \boxed{4} \).
To check, we can verify how many soaps and bottles go into each basket when we divide the items accordingly:
- If she makes 4 baskets, then she would put:
- \( \frac{16 \text{ soaps}}{4 \text{ baskets}} = 4 \text{ soaps per basket} \)
- \( \frac{12 \text{ bottles}}{4 \text{ baskets}} = 3 \text{ bottles per basket} \)
This confirms that each of the 4 baskets can have 4 soaps and 3 bottles of bubble bath, utilizing all items without any remainder.