find the GCF and LCM of 84 and 96?
The correct answer and explanation is:
To find the Greatest Common Factor (GCF) and the Least Common Multiple (LCM) of 84 and 96, we follow these steps:
Step 1: Prime Factorization
First, we need the prime factorizations of the two numbers.
- For 84:
84 is divisible by 2 (since it is even), so: 84÷2=4284 \div 2 = 42 42 is also divisible by 2: 42÷2=2142 \div 2 = 21 21 is divisible by 3: 21÷3=721 \div 3 = 7 7 is a prime number, so the prime factorization of 84 is: 84=22×3×784 = 2^2 \times 3 \times 7 - For 96:
96 is divisible by 2: 96÷2=4896 \div 2 = 48 48 is divisible by 2: 48÷2=2448 \div 2 = 24 24 is divisible by 2: 24÷2=1224 \div 2 = 12 12 is divisible by 2: 12÷2=612 \div 2 = 6 6 is divisible by 2: 6÷2=36 \div 2 = 3 3 is a prime number, so the prime factorization of 96 is: 96=25×396 = 2^5 \times 3
Step 2: Find the GCF
The GCF is found by taking the lowest power of each prime factor that appears in both factorizations. From the prime factorizations:
- For 2, the minimum exponent is 2 (from 84, where it is 222^2).
- For 3, the minimum exponent is 1 (appears in both 84 and 96).
- For 7, it only appears in 84, so it is not included in the GCF.
Thus, the GCF is: GCF=22×3=4×3=12\text{GCF} = 2^2 \times 3 = 4 \times 3 = 12
Step 3: Find the LCM
The LCM is found by taking the highest power of each prime factor that appears in either number’s factorization. From the prime factorizations:
- For 2, the highest exponent is 5 (from 96, where it is 252^5).
- For 3, the highest exponent is 1 (appears in both).
- For 7, the highest exponent is 1 (from 84).
Thus, the LCM is: LCM=25×3×7=32×3×7=672\text{LCM} = 2^5 \times 3 \times 7 = 32 \times 3 \times 7 = 672
Final Answer:
- GCF of 84 and 96 = 12
- LCM of 84 and 96 = 672
Explanation:
The GCF and LCM of two numbers are useful for solving problems involving fractions, ratios, and common multiples. The GCF represents the largest number that divides both numbers without a remainder, while the LCM represents the smallest number that both numbers divide evenly into. Using prime factorization allows us to systematically identify the common and unique factors that help determine these values.