Question

find the gcf for the following list.

12, 28, 56

the gcf is (simplify your answer.)

Explanation:

🆕 New Concept Discovered: Greatest Common Factor (GCF)
The largest number that divides evenly into a set of numbers.

Step 1: Find prime factorizations

Find the prime factors for each number in the list:

$$ 12 = 2 \times 2 \times 3 = 2^2 \times 3 $$

$$ 28 = 2 \times 2 \times 7 = 2^2 \times 7 $$

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

Step 2: Identify common prime factors

Identify the prime factors that appear in all three factorizations and choose the lowest power of each:

• The prime factor \(2\) is common to all three lists.
• The lowest power of \(2\) among the lists is \(2^2\).
• The prime factors \(3\) and \(7\) are not common to all three numbers.

Step 3: Calculate the GCF

Multiply the common prime factors with their lowest powers:

$$ \text{GCF} = 2^2 = 4 $$

Answer:

4