Question

find the gcf for the given list.

32, 56

the gcf is

Explanation:

🆕 New Concept Discovered: Greatest Common Factor (GCF)
Finding the largest shared divisor of two numbers

Step 1: Find the prime factorization of 32

To find the Greatest Common Factor (GCF), we first break down each number into its basic building blocks, which are prime numbers.

Let's divide \( 32 \) by its smallest prime factor, \( 2 \):

$$ 32 = 2 \times 16 $$

$$ 16 = 2 \times 8 $$

$$ 8 = 2 \times 4 $$

$$ 4 = 2 \times 2 $$

So, the prime factorization of \( 32 \) is:

$$ 32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5 $$

Step 2: Find the prime factorization of 56

Now, let's break down \( 56 \) into its prime factors:

$$ 56 = 2 \times 28 $$

$$ 28 = 2 \times 14 $$

$$ 14 = 2 \times 7 $$

Since \( 7 \) is a prime number, the prime factorization of \( 56 \) is:

$$ 56 = 2 \times 2 \times 2 \times 7 = 2^3 \times 7 $$

Step 3: Identify the common prime factors

To find the GCF, we look for the prime factors that both numbers share and choose the lowest power of those shared factors.

• Both numbers share the prime factor \( 2 \).
• The power of \( 2 \) in \( 32 \) is \( 2^5 \).
• The power of \( 2 \) in \( 56 \) is \( 2^3 \).

We take the lowest exponent, which is \( 2^3 \):

$$ \text{GCF} = 2^3 = 2 \times 2 \times 2 = 8 $$

Answer:

8