Explain how to use prime factorization to find the GCF of 90 and 84 without drawing Venn diagram: How is the symbolic method similar to and different from the Venn diagram method of= ‘finding the GCF?
The Correct Answer and Explanation is:
To find the GCF (Greatest Common Factor) of 90 and 84 using prime factorization (without using a Venn diagram), follow these steps:
Step 1: Prime factorization of both numbers
90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
So, the prime factorization of 90 is:
90 = 2 × 3 × 3 × 5 or 2 × 3² × 5
84
84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1
So, the prime factorization of 84 is:
84 = 2 × 2 × 3 × 7 or 2² × 3 × 7
Step 2: Identify common prime factors
Look for the prime numbers that appear in both factorizations.
- Both numbers have 2: 90 has one 2, and 84 has two 2s → take the lowest power, which is one 2.
- Both numbers have 3: 90 has two 3s, and 84 has one 3 → take the lowest power, which is one 3.
- 5 appears only in 90, and 7 appears only in 84 → do not include them.
Step 3: Multiply the common prime factors
GCF = 2 × 3 = 6
Explanation of Symbolic vs. Venn Diagram Method
Both methods involve finding common prime factors. The symbolic method uses multiplication and exponents to list prime factors and compare powers. The Venn diagram method uses two circles to visually group shared and unshared primes. The overlapping part of the diagram shows common primes used to find the GCF.
The similarity is that both methods rely on identifying common primes and multiplying them to get the GCF. The difference is in the presentation: the symbolic method is more numerical and abstract, while the Venn diagram is visual and can be easier for learners who prefer diagrams. Both reach the same answer.
