What is the HCF of 84, 120, 138 in the prime factorization method?
The correct answer and explanation is:
To find the Highest Common Factor (HCF) of 84, 120, and 138 using the prime factorization method, we first need to break down each number into its prime factors.
- Prime factorization of 84:
84 is an even number, so we start by dividing by 2: 84÷2=4284 \div 2 = 42 42 is also even, so divide by 2 again: 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 - Prime factorization of 120:
120 is an even number, so divide by 2: 120÷2=60120 \div 2 = 60 60 is also even, so divide by 2 again: 60÷2=3060 \div 2 = 30 30 is even, so divide by 2 one more time: 30÷2=1530 \div 2 = 15 15 is divisible by 3: 15÷3=515 \div 3 = 5 5 is a prime number. So, the prime factorization of 120 is: 120=23×3×5120 = 2^3 \times 3 \times 5 - Prime factorization of 138:
138 is an even number, so divide by 2: 138÷2=69138 \div 2 = 69 69 is divisible by 3: 69÷3=2369 \div 3 = 23 23 is a prime number. So, the prime factorization of 138 is: 138=2×3×23138 = 2 \times 3 \times 23
Now that we have the prime factorizations of all three numbers:
- 84 = 22×3×72^2 \times 3 \times 7
- 120 = 23×3×52^3 \times 3 \times 5
- 138 = 2×3×232 \times 3 \times 23
The next step is to find the common prime factors. The common prime factors in all three numbers are 2 and 3.
- The lowest power of 2 common to all three numbers is 212^1.
- The lowest power of 3 common to all three numbers is 313^1.
Thus, the HCF is: HCF=21×31=2×3=6HCF = 2^1 \times 3^1 = 2 \times 3 = 6
So, the HCF of 84, 120, and 138 is 6.