Finding GCF by Listing Factors

Find the GCF of $24$ and $36$.

List all factors of 24: $24 = 1 \times 24 = 2 \times 12 = 3 \times 8 = 4 \times 6$ = Factors: $1, 2, 3, 4, 6, 8, 12, 24$

List all factors of 36: $36 = 1 \times 36 = 2 \times 18 = 3 \times 12 = 4 \times 9 = 6 \times 6$ = Factors: $1, 2, 3, 4, 6, 9, 12, 18, 36$

Circle the common factors: Both lists contain: $1, 2, 3, 4, 6, 12$ = Common factors found

Find the greatest: The largest number in both lists is $12$ = $\text{GCF}(24, 36) = 12$

Answer: The GCF of 24 and 36 is $12$.

Finding GCF Using Prime Factorization

Find the GCF of $48$ and $60$ using prime factorization.

Find prime factorization of 48: $48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3$ = $48 = 2^4 \times 3$

Find prime factorization of 60: $60 = 2 \times 30 = 2 \times 2 \times 15 = 2 \times 2 \times 3 \times 5$ = $60 = 2^2 \times 3 \times 5$

Identify common prime factors: Both have: $2$ (appears in both) and $3$ (appears in both) = Common primes: $2$ and $3$

Take lowest power of each common prime: $2$: lowest power is $2^2$; $3$: lowest power is $3^1$ = Use $2^2$ and $3^1$

Multiply the common factors: $2^2 \times 3 = 4 \times 3 = 12$ = $\text{GCF}(48, 60) = 12$

Answer: The GCF of 48 and 60 is $12$.

Finding GCF of Three Numbers

Find the GCF of $18$, $24$, and $30$.

Prime factorization of 18: $18 = 2 \times 9 = 2 \times 3 \times 3$ = $18 = 2 \times 3^2$

Prime factorization of 24: $24 = 2 \times 12 = 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 3$ = $24 = 2^3 \times 3$

Prime factorization of 30: $30 = 2 \times 15 = 2 \times 3 \times 5$ = $30 = 2 \times 3 \times 5$

Find primes common to ALL three: $2$ appears in all (lowest: $2^1$); $3$ appears in all (lowest: $3^1$) = Common: $2^1$ and $3^1$

Multiply common factors: $2 \times 3 = 6$ = $\text{GCF}(18, 24, 30) = 6$

Answer: The GCF of 18, 24, and 30 is $6$.

Using GCF to Simplify Fractions

Simplify $\frac{36}{48}$ using the GCF.

Find GCF of numerator and denominator: Factors of 36: $1, 2, 3, 4, 6, 9, 12, 18, 36$; Factors of 48: $1, 2, 3, 4, 6, 8, 12, 16, 24, 48$ = $\text{GCF}(36, 48) = 12$

Divide both by the GCF: $\frac{36 \div 12}{48 \div 12} = \frac{3}{4}$ = Simplified fraction

Verify it's fully simplified: $\text{GCF}(3, 4) = 1$, so it's in lowest terms = $\frac{3}{4}$ is simplified

Answer: $\frac{36}{48} = \frac{3}{4}$

Mistake: Confusing GCF with LCM

Why: GCF is about dividing (finding what goes INTO both numbers), while LCM is about multiplying (finding what both numbers go INTO).

Correct: GCF of 12 and 18 is 6 (divides both). LCM of 12 and 18 is 36 (both divide into it).

Mistake: Forgetting to take the LOWEST power when using prime factorization

Why: Students sometimes multiply all prime factors instead of only the common ones with lowest powers.

Correct: For $2^4 \times 3$ and $2^2 \times 3 \times 5$, use $2^2$ (not $2^4$) and $3^1$. GCF = $2^2 \times 3 = 12$.

Mistake: Stopping before finding the GREATEST common factor

Why: Finding any common factor doesn't mean it's the greatest one.

Correct: For 24 and 36, common factors are 1, 2, 3, 4, 6, 12. The GCF is 12, not just any common factor like 2 or 6.

Simplifying Fractions

Reducing fractions to lowest terms requires finding the GCF of the numerator and denominator.

To simplify $\frac{42}{56}$: GCF(42, 56) = 14, so $\frac{42}{56} = \frac{42 \div 14}{56 \div 14} = \frac{3}{4}$

Cutting Materials Without Waste

When cutting boards or fabric, the GCF helps find the largest equal pieces possible.

You have a 72 cm board and a 90 cm board. To cut them into equal pieces with no waste, use pieces of GCF(72, 90) = 18 cm.

Organizing Items into Groups

The GCF helps when distributing different types of items into equal groups.

With 28 apples and 42 oranges, you can make GCF(28, 42) = 14 identical gift baskets, each with 2 apples and 3 oranges.

Q: What if the GCF of two numbers is 1?

A: When GCF = 1, the numbers are called **relatively prime** or **coprime**. They share no common factors other than 1. Examples: GCF(8, 15) = 1, GCF(7, 12) = 1.

Q: Can the GCF be larger than one of the numbers?

A: No, the GCF is always less than or equal to the smaller number. The GCF must divide both numbers, so it can't exceed either of them.

Q: How is GCF different from LCM?

A: GCF is the largest number that divides INTO both numbers. LCM (Least Common Multiple) is the smallest number that both numbers divide INTO. For 12 and 18: GCF = 6, LCM = 36.