Find the product of 12 and 8. Find the greatest common factor of 12 and 8. Find the least common multiple of 12 and 8. Find the product of the greatest common factor and the least common multiple of 12 and 8.
The Correct Answer and Explanation is :
To solve the given problems regarding the numbers 12 and 8, we will find the product, greatest common factor (GCF), least common multiple (LCM), and then compute the product of the GCF and LCM.
1. Product of 12 and 8
The product of two numbers is calculated by multiplying them:
[
\text{Product} = 12 \times 8 = 96
]
2. Greatest Common Factor (GCF) of 12 and 8
The GCF is the largest number that divides both 12 and 8 without leaving a remainder. To find the GCF, we can list the factors of each number:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 8: 1, 2, 4, 8
The common factors are 1, 2, and 4. The greatest of these is:
[
\text{GCF} = 4
]
3. Least Common Multiple (LCM) of 12 and 8
The LCM is the smallest number that both 12 and 8 can divide without a remainder. We can find the LCM using the relationship between GCF and LCM, which is:
[
\text{LCM} = \frac{\text{Product}}{\text{GCF}}
]
Substituting the values we calculated:
[
\text{LCM} = \frac{96}{4} = 24
]
4. Product of GCF and LCM
Now, we find the product of the GCF and LCM:
[
\text{Product of GCF and LCM} = 4 \times 24 = 96
]
Summary of Results
- Product of 12 and 8: 96
- Greatest Common Factor of 12 and 8: 4
- Least Common Multiple of 12 and 8: 24
- Product of GCF and LCM: 96
Explanation
In this process, we demonstrated the relationships between multiplication, common factors, and multiples. The product of two numbers (96) is also equal to the product of their GCF (4) and LCM (24). This illustrates a fundamental property in number theory, showing how GCF and LCM encapsulate the structure of integers and their divisibility. Understanding these concepts helps in various applications, including simplifying fractions, solving problems in algebra, and managing data in computer science. Recognizing the GCF and LCM is essential for factoring polynomials, finding equivalent fractions, and performing operations on fractions efficiently.