Understanding Common Factors: Guide and Examples for Grades 4-7

What Are Common Factors?

When comparing the factors of two or more numbers, you will often find that they share some of the same divisors. These shared divisors are called common factors.

If a number divides exactly into every number in your given set, it belongs to the list of common factors for that set.

A common factor divides a pair of two or more numbers exactly, leaving a remainder of zero.

For example, the numbers $1, 2, 4$, and $8$ are all factors of $8$. The numbers $1, 2, 3, 4, 6$, and $12$ are all factors of $12$. The numbers that appear in both groups are $1, 2$, and $4$. Therefore, $1, 2$, and $4$ are the common factors of $8$ and $12$.

Key Ideas and Vocabulary

Every whole number greater than $1$ has at least two factors: $1$ and itself. Because $1$ divides into every whole number without a remainder, $1$ is a common factor for any set of numbers you choose.

If $1$ is the only shared factor between two numbers, those numbers are called co-prime. For example, the factors of $9$ are $1, 3$, and $9$. The factors of $10$ are $1, 2, 5$, and $10$. They share no divisors other than $1$, making $9$ and $10$ co-prime numbers.

The largest shared number in a set of common factors is called the greatest common factor. Identifying the largest shared divisor helps solve problems involving maximum equal groupings.

Visual Explanation

A Venn diagram is an excellent way to visualize how factors relate. By placing the factors of one number in the left circle and the factors of another number in the right circle, the shared intersection in the middle clearly displays the common factors.

The overlap in the center collects the divisors that work for both $12$ and $18$. By isolating the intersection, you can easily identify that the shared factors are $1, 2, 3$, and $6$.

Worked Examples

Finding shared factors is straightforward when you use systematic lists or factorization. Review the examples below to see different methods applied.

Example 1: Finding shared factors using lists

Question: What are the common factors of $18$ and $27$?

Method:

1. List all factors of $18$ by finding dividing pairs: $1, 2, 3, 6, 9, 18$.
2. List all factors of $27$: $1, 3, 9, 27$.
3. Identify the numbers that appear in both factor lists.

Answer: The common factors are $1, 3$, and $9$.

Check: Verify that both $18$ and $27$ can be divided by $9$ without leaving a remainder. $18 \div 9 = 2$ and $27 \div 9 = 3$.

Example 2: Finding shared factors for three numbers

Question: What are the common factors of $12, 30$, and $42$?

Method:

1. Write out the full list of factors for $12$: $1, 2, 3, 4, 6, 12$.
2. Write out the full list of factors for $30$: $1, 2, 3, 5, 6, 10, 15, 30$.
3. Write out the full list of factors for $42$: $1, 2, 3, 6, 7, 14, 21, 42$.
4. Locate the specific numbers that are present in all three lists.

Answer: The common factors are $1, 2, 3$, and $6$.

Check: The largest shared divisor is $6$. Since $12 \div 6 = 2$, $30 \div 6 = 5$, and $42 \div 6 = 7$, the factors are correct.

Example 3: Using prime factorization

Question: Find the greatest common factor of $36$ and $60$ using prime factors.

Method:

1. Use a factor tree to find the prime factorization of $36$. This breaks down into $2 \times 2 \times 3 \times 3$.
2. Find the prime factorization of $60$. This breaks down into $2 \times 2 \times 3 \times 5$.
3. Identify the prime numbers shared by both lists. They both contain two $2$s and one $3$.
4. Multiply the shared prime numbers together: $2 \times 2 \times 3$.

Answer: The greatest common factor is $12$.

Check: Divide both original numbers by $12$. We get $36 \div 12 = 3$ and $60 \div 12 = 5$. Because $3$ and $5$ share no other factors, $12$ is accurately confirmed as the greatest shared divisor.

Common Mistakes and Non-Examples

A frequent error is confusing shared factors with common multiples. Factors are divisors that are equal to or smaller than the given number. Multiples are the result of multiplying the number, making them equal to or larger than the given number.

Always check every proposed factor to ensure it leaves no remainder for both numbers. For example, a student might guess that $4$ is a common factor of $12$ and $18$ because it divides evenly into $12$. However, $4$ is a non-example here. When you divide $18$ by $4$, it leaves a remainder of $2$. Therefore, $4$ cannot be shared by both.

Real-World Connections

You rely on common factors whenever you need to organize different items into equal groups.

Suppose you have $24$ apples and $32$ oranges, and you want to pack them into identical gift baskets with no fruit left over. Finding the common factors of $24$ and $32$ tells you exactly how many baskets you can make. Since their shared factors are $1, 2, 4$, and $8$, you have the option of making $1, 2, 4$, or $8$ identical baskets containing both types of fruit.