To find the greatest number of centerpieces Emily can make while using all of her balloons with the same number of each color balloon per centerpiece, we need to determine the greatest common divisor (GCD) of the numbers of yellow and green balloons.
First, we find the GCD of 24 and 32 by applying the Euclidean algorithm.
**Step-by-step application of the Euclidean algorithm:**
1. **Divide 32 by 24**:
\[
32 \div 24 = 1 \quad \text{with a remainder of} \quad 32 - 1 \times 24 = 8
\]
2. **Now, divide 24 by the remainder 8**:
\[
24 \div 8 = 3 \quad \text{with a remainder of} \quad 24 - 3 \times 8 = 0
\]
Since the remainder is now 0, the last non-zero remainder (which is 8) is the GCD of 24 and 32.
**Confirming the GCD through prime factorization:**
- \(24\) can be factored into prime factors:
\[
24 = 2^3 \times 3
\]
- \(32\) can be factored into prime factors:
\[
32 = 2^5
\]
The common prime factor is \(2\). The smallest power of \(2\) common to both factorizations is \(2^3\).
Thus:
\[
\text{GCD} = 2^3 = 8
\]
**Conclusion:**
The greatest number of centerpieces Emily can make, while ensuring each centerpiece has the same number of yellow and green balloons and using all balloons, is given by the GCD of 24 and 32, which is \(8\).
Thus, Emily can make a maximum of \( \boxed{8} \) centerpieces.
Emily is making balloons centerpieces to decorate for the spring dance. He has 24 yellow balloons and 32 green balloons. She wants to have the same number of each color balloon every centerpiece. If she uses all of the balloons, what is the greatest number of centerpieces she can make?
3 answers
Antonio is making sandwiches for his soccer team year and party. He has 18 slices of cheese and 27 slices of turkey cheese for the sandwiches. He wants to make identical sandwiches that use all the slices. What is the greatest sandwiches Antonio can make?
To determine the greatest number of identical sandwiches Antonio can make using all the slices of cheese and turkey, we need to find the greatest common divisor (GCD) of the numbers of cheese slices and turkey slices.
Antonio has:
- 18 slices of cheese
- 27 slices of turkey
We will find the GCD of 18 and 27 using the Euclidean algorithm.
**Step-by-step application of the Euclidean algorithm:**
1. **Divide 27 by 18**:
\[
27 \div 18 = 1 \quad \text{with a remainder of} \quad 27 - 1 \times 18 = 9
\]
2. **Now, divide 18 by the remainder 9**:
\[
18 \div 9 = 2 \quad \text{with a remainder of} \quad 18 - 2 \times 9 = 0
\]
Since the remainder is now 0, the last non-zero remainder (which is 9) is the GCD of 18 and 27.
**Confirming the GCD through prime factorization:**
- \(18\) can be factored into prime factors:
\[
18 = 2 \times 3^2
\]
- \(27\) can be factored into prime factors:
\[
27 = 3^3
\]
The common prime factor is \(3\). The smallest power of \(3\) common to both factorizations is \(3^2\).
Thus:
\[
\text{GCD} = 3^2 = 9
\]
**Conclusion:**
The greatest number of sandwiches Antonio can make, while ensuring each sandwich uses the same number of cheese and turkey slices and uses all of them, is given by the GCD of 18 and 27, which is \(9\).
Thus, Antonio can make a maximum of \( \boxed{9} \) identical sandwiches.
Antonio has:
- 18 slices of cheese
- 27 slices of turkey
We will find the GCD of 18 and 27 using the Euclidean algorithm.
**Step-by-step application of the Euclidean algorithm:**
1. **Divide 27 by 18**:
\[
27 \div 18 = 1 \quad \text{with a remainder of} \quad 27 - 1 \times 18 = 9
\]
2. **Now, divide 18 by the remainder 9**:
\[
18 \div 9 = 2 \quad \text{with a remainder of} \quad 18 - 2 \times 9 = 0
\]
Since the remainder is now 0, the last non-zero remainder (which is 9) is the GCD of 18 and 27.
**Confirming the GCD through prime factorization:**
- \(18\) can be factored into prime factors:
\[
18 = 2 \times 3^2
\]
- \(27\) can be factored into prime factors:
\[
27 = 3^3
\]
The common prime factor is \(3\). The smallest power of \(3\) common to both factorizations is \(3^2\).
Thus:
\[
\text{GCD} = 3^2 = 9
\]
**Conclusion:**
The greatest number of sandwiches Antonio can make, while ensuring each sandwich uses the same number of cheese and turkey slices and uses all of them, is given by the GCD of 18 and 27, which is \(9\).
Thus, Antonio can make a maximum of \( \boxed{9} \) identical sandwiches.