How can I find the HCF of 1008 and 1080 using the prime factorization method?

To find the Highest Common Factor (HCF) of 1008 and 1080 using the prime factorization method, we will follow these steps:

1. Prime Factorization of 1008:
• Divide 1008 by the smallest prime number, which is 2. 1008 ÷ 2 = 504
• Continue dividing by 2: 504 ÷ 2 = 252
• Again: 252 ÷ 2 = 126
• And again: 126 ÷ 2 = 63
• Now, 63 can no longer be divided by 2, so we move to the next prime number, which is 3: 63 ÷ 3 = 21
• Finally: 21 ÷ 3 = 7
• Since 7 is a prime number, we stop here.

Thus, the prime factorization of 1008 is:

1008 = 2⁴ × 3² × 7

2. Prime Factorization of 1080:
• Again, we start by dividing by 2: 1080 ÷ 2 = 540
• Continue: 540 ÷ 2 = 270
• Next: 270 ÷ 2 = 135
• Now we switch to 3, as 135 is not divisible by 2: 135 ÷ 3 = 45
• Next: 45 ÷ 3 = 15
• Finally, 15 ÷ 3 = 5
• 5 is prime, so we stop here.

Thus, the prime factorization of 1080 is:

1080 = 2³ × 3³ × 5

3. Finding the HCF:

Now that we have the prime factorizations:

• 1008 = 2⁴ × 3² × 7
• 1080 = 2³ × 3³ × 5

We identify the common prime factors and take the lowest powers:

• For the prime factor 2: The lowest power is 2³
• For the prime factor 3: The lowest power is 3²
• The prime factors 5 and 7 are not common, so they are ignored.

Now, we calculate the HCF:

HCF = 2³ × 3² = 8 × 9 = 72

Thus, the HCF of 1008 and 1080 using the prime factorization method is 72.