Find the HCF of 45 , 60 and 72 using prime factorization”
The correct answer and explanation is:
To find the Highest Common Factor (HCF) of 45, 60, and 72 using prime factorization, follow these steps:
Step 1: Find the prime factorization of each number.
- Prime factorization of 45:
- Start by dividing 45 by 3 (the smallest prime number):
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
- Prime factorization of 45: $45 = 3^2 \times 5$
- Prime factorization of 60:
- Start by dividing 60 by 2 (the smallest prime number):
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
- Prime factorization of 60: $60 = 2^2 \times 3 \times 5$
- Prime factorization of 72:
- Start by dividing 72 by 2 (the smallest prime number):
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- Prime factorization of 72: $72 = 2^3 \times 3^2$
Step 2: Find the common prime factors.
Now, compare the prime factorizations of all three numbers:
- $45 = 3^2 \times 5$
- $60 = 2^2 \times 3 \times 5$
- $72 = 2^3 \times 3^2$
The only common prime factor between all three numbers is 3.
Step 3: Choose the lowest power of the common prime factors.
For 3, the lowest power is $3^1$, because it appears as $3^1$ in 60 and $3^2$ in both 45 and 72.
Step 4: Calculate the HCF.
Since 3 is the only common prime factor, the HCF of 45, 60, and 72 is $3^1 = 3$.
Conclusion:
The HCF of 45, 60, and 72 is 3. This method ensures that we only consider the prime factors common to all three numbers and select the smallest power for each prime factor to get the highest common divisor.