5-5 Factors and Multiples

Property

• A factor is a whole number that divides exactly into another number. If $a \times b = c$, then $a$ and $b$ are factors of $c$.
• A multiple of a number is the result of multiplying that number by a counting number (1, 2, 3, ...).

Examples

Property

To find the Greatest Common Factor ($\text{GCF}$) of two or more numbers by listing:

1. List all the factors of each number.
2. Identify the common factors shared by all the numbers.
3. Identify the greatest number from the list of common factors.

Examples

Property

To find the GCF of two or more numbers:
Step 1: List the prime factors for each number.
Step 2: Identify the shared factors.
Step 3: Multiply the shared factors to find the GCF.

Examples

• For $36$ and $60$: $36 = 2 \cdot 2 \cdot 3 \cdot 3$ and $60 = 2 \cdot 2 \cdot 3 \cdot 5$. They share two $2$s and one $3$. The GCF is $2 \cdot 2 \cdot 3 = 12$.
• For $18$ and $81$: $18 = 2 \cdot 3 \cdot 3$ and $81 = 3 \cdot 3 \cdot 3 \cdot 3$. They share two $3$s. The GCF is $3 \cdot 3 = 9$.

Explanation

Finding the GCF is like looking at two friends' collections of trading cards (their prime factors). You pull out all the cards they have in common—the shared factors. The combined value of these shared cards is the Greatest Common Factor! It’s the biggest number that can be built from the ingredients that both numbers share in their prime factorization recipes.