Factor Forge Notes Worksheet
FACTOR FORGE
GCF via Prime Factor Pairs
Finding the Greatest Common Factor Using Factor Trees
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Concept & Method: Finding GCF from Prime Factor Pairs
Greatest Common Factor (GCF):
The largest factor that two or more numbers have in common.
Why Prime Factorization?
Listing factors can easily miss large numbers. Multiplying shared prime factors guarantees finding the exact GCF every time!
If two numbers share no common prime factors, their GCF is 1.
The 3-Step Strategy:
- Factor Both: Draw a factor tree for each number in the pair.
- Match Pairs: Identify every prime factor shared by both prime lists.
- Multiply: Multiply only the matching primes together to get the GCF.
\(\text{GCF} = \text{Product of all shared primes}\)
Worked Example: GCF(24, 36)
\(24 = \) 2 × 2 × 2 × 3
\(36 = \) 2 × 2 × 3 × 3
Shared: \(2, 2, 3\) \(\text{GCF} = 2 \times 2 \times 3 = 12\)
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Practice Problems: Prime Factor Pairs to Find GCF
Construct both factor trees, identify shared prime factors, and multiply to find the GCF.
Problem 1: Pair (18, 24) Factor Tree Workspaces
Tree for 18:
Tree for 24:
Shared Primes:
GCF(18, 24) =
Problem 2: Pair (30, 45) Factor Tree Workspaces
Tree for 30:
Tree for 45:
Shared Primes:
GCF(30, 45) =
Problem 3: Pair (40, 60) Factor Tree Workspaces
Tree for 40:
Tree for 60:
Shared Primes:
GCF(40, 60) =
Problem 4: Pair (28, 70) Factor Tree Workspaces
Tree for 28:
Tree for 70:
Shared Primes:
GCF(28, 70) =
Factor Forge Answer Key
FACTOR FORGE
Teacher Answer Key
Complete Factor Trees, Prime Breakdowns & GCF Solutions
CCSS.MATH.6.NS.B.4 100% Verified Key
Teacher Notes & Common Misconceptions
Misconception 1: Not Multiplying
Students often list shared factors (e.g., "2 and 3") but forget to multiply them together to reach the final GCF (6).
Misconception 2: Multiple Branches
Remind students factor trees can start with different pairs (e.g., 24 = 2×12 or 4×6), but the final prime leaves are always identical.
Quick Check Method
Verify by dividing both original numbers by the GCF. The resulting quotients must share no common factors other than 1.
KEY
Step-by-Step Problem Solutions
Circled numbers indicate prime leaves
Problem 1: Pair (18, 24) GCF = 6
18 / \
2 9
/ \
3 3
24 / \
4 6
/ \ / \
2 2 2 3
18 = 2 × 3 × 3 24 = 2 × 2 × 2 × 3
Shared Primes: 2, 3 GCF = 2 × 3 = 6
Problem 2: Pair (30, 45) GCF = 15
30 / \
3 10
/ \
2 5
45 / \
5 9
/ \
3 3
30 = 2 × 3 × 5 45 = 3 × 3 × 5
Shared Primes: 3, 5 GCF = 3 × 5 = 15
Problem 3: Pair (40, 60) GCF = 20
40 / \
4 10
/ \ / \
2 2 2 5
60 / \
6 10
/ \ / \
2 3 2 5
40 = 2 × 2 × 2 × 5 60 = 2 × 2 × 3 × 5
Shared Primes: 2, 2, 5 GCF = 2 × 2 × 5 = 20
Problem 4: Pair (28, 70) GCF = 14
28 / \
4 7
/ \
2 2
70 / \
7 10
/ \
2 5
28 = 2 × 2 × 7 70 = 2 × 5 × 7
Shared Primes: 2, 7 GCF = 2 × 7 = 14