To find the Highest Common Factor (HCF) of 1008 and 1080 using the prime factorization method, we first need to break down both numbers into their prime factors.
Step 1: Prime Factorization of 1008
We start with 1008:
- 1008 is even, so we start dividing by 2.
- 1008 ÷ 2 = 504
- 504 ÷ 2 = 252
- 252 ÷ 2 = 126
- 126 ÷ 2 = 63
- 63 is not even but divisible by 3: 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- 7 is a prime number.
So the prime factorization of 1008 is: 24 × 32 × 7
Step 2: Prime Factorization of 1080
Next, we do prime factorization for 1080:
- 1080 is even, divide by 2.
- 1080 ÷ 2 = 540
- 540 ÷ 2 = 270
- 270 ÷ 2 = 135
- 135 is not even but divisible by 3: 135 ÷ 3 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 is a prime number.
So the prime factorization of 1080 is: 23 × 33 × 5
Step 3: Identify Common Factors
We now list the prime factors of both numbers:
- 1008: 24 × 32 × 7
- 1080: 23 × 33 × 5
The common prime factors are:
- 2 (minimum power is 23)
- 3 (minimum power is 32)
Step 4: Calculate the HCF
Now we calculate the HCF by taking the lowest power of all common prime factors:
HCF = 23 × 32 = 8 × 9 = 72
Conclusion
The HCF of 1008 and 1080 is 72.