Find the HCF of the following numbers by prime factorization method 180, 270. 630
The correct answer and explanation is:
To find the highest common factor (HCF) of the numbers 180, 270, and 630 using the prime factorization method, we will follow these steps:
Step 1: Prime Factorization of Each Number
Start by finding the prime factorization of each number.
180:
- 180 is divisible by 2:
180 ÷ 2 = 90 - 90 is divisible by 2:
90 ÷ 2 = 45 - 45 is divisible by 3:
45 ÷ 3 = 15 - 15 is divisible by 3:
15 ÷ 3 = 5 - 5 is prime.
Thus, the prime factorization of 180 is:
180=22×32×5180 = 2^2 \times 3^2 \times 5
270:
- 270 is divisible by 2:
270 ÷ 2 = 135 - 135 is divisible by 3:
135 ÷ 3 = 45 - 45 is divisible by 3:
45 ÷ 3 = 15 - 15 is divisible by 3:
15 ÷ 3 = 5 - 5 is prime.
Thus, the prime factorization of 270 is:
270=2×33×5270 = 2 \times 3^3 \times 5
630:
- 630 is divisible by 2:
630 ÷ 2 = 315 - 315 is divisible by 3:
315 ÷ 3 = 105 - 105 is divisible by 3:
105 ÷ 3 = 35 - 35 is divisible by 5:
35 ÷ 5 = 7 - 7 is prime.
Thus, the prime factorization of 630 is:
630=2×32×5×7630 = 2 \times 3^2 \times 5 \times 7
Step 2: Identify the Common Factors
Now, identify the common factors from the prime factorizations of all three numbers:
- For 2: The lowest power is 212^1.
- For 3: The lowest power is 323^2.
- For 5: The lowest power is 515^1.
Step 3: Multiply the Common Prime Factors
Now, multiply the common prime factors: HCF=21×32×51=2×9×5=90HCF = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90
Final Answer:
Thus, the HCF of 180, 270, and 630 is 90.
Explanation:
The prime factorization method is an efficient way to find the HCF because it breaks down each number into its prime factors. Once the numbers are expressed as products of prime numbers, finding the HCF is simply a matter of identifying the smallest powers of the common prime factors and multiplying them. In this case, the common factors are 2, 3, and 5, and by multiplying their lowest powers, we get 90 as the highest common factor.