What is the gcf of 96×5 and 64×2 – In the realm of mathematics, where numbers dance and equations unravel secrets, we embark on a quest to uncover the Greatest Common Factor (GCF) of two enigmatic expressions: 96x^5 and 64x^2. This exploration will lead us through the labyrinth of prime factorization, unveil the intricacies of common factors, and culminate in the triumphant revelation of their GCF.
Greatest Common Factor (GCF)
The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the given expressions.
For example, the GCF of 12 and 18 is 6, since 6 is the largest number that divides both 12 and 18 evenly.
Prime Factorization
Prime factorization is a method for finding the GCF of two or more expressions by breaking them down into their prime factors.
To find the prime factorization of a number, you divide the number by the smallest prime number that divides it evenly. You then continue dividing the quotient by the next smallest prime number that divides it evenly, and so on.
You stop when you reach a quotient that is prime.
For example, the prime factorization of 12 is 2 x 2 x 3, and the prime factorization of 18 is 2 x 3 x 3.
Identifying Common Factors
To find the GCF of two or more expressions, you first find the prime factorization of each expression. Then, you identify the common factors that appear in all of the prime factorizations.
For example, the common factors of 12 and 18 are 2 and 3.
| Expression | Prime Factorization | Common Factors |
|---|---|---|
| 12 | 2 x 2 x 3 | 2, 3 |
| 18 | 2 x 3 x 3 | 2, 3 |
Calculating the GCF, What is the gcf of 96×5 and 64×2
Once you have identified the common factors, you can calculate the GCF by multiplying the common factors together.
For example, the GCF of 12 and 18 is 2 x 3 = 6.
Real-World Applications
The GCF has many applications in the real world, including:
- Engineering: The GCF can be used to find the greatest common divisor of two or more numbers, which is useful for designing structures that can withstand multiple loads.
- Mathematics: The GCF can be used to simplify fractions and solve equations.
- Everyday problem-solving: The GCF can be used to find the greatest common factor of two or more quantities, which can be useful for solving problems such as finding the greatest common divisor of two or more numbers.
FAQ Section: What Is The Gcf Of 96×5 And 64×2
What is the GCF of 96x^5 and 64x^2?
The GCF of 96x^5 and 64x^2 is 64x^2.
How do you find the GCF of two expressions?
To find the GCF, prime factorize each expression, identify the common factors, and multiply their lowest exponents.
