Find the HCF of the following numbers by prime factorization method.
(a)24,60 (b)48,64,80

Explanation

To find the HCF (Highest Common Factor) of the given numbers using the prime factorization method, we will follow these steps for both parts of the question.

### Part (a): Find the HCF of 24 and 60

**Step 1: Prime Factorization of 24**
- Divide 24 by the smallest prime number, which is 2:
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 is a prime number, so we stop here.
- The prime factorization of 24 is:
- \( 24 = 2^3 \times 3^1 \)

**Step 2: Prime Factorization of 60**
- Divide 60 by the smallest prime number, which is 2:
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5
- 5 is a prime number, so we stop here.
- The prime factorization of 60 is:
- \( 60 = 2^2 \times 3^1 \times 5^1 \)

**Step 3: Identify Common Factors**
- Now, we identify the common prime factors from both factorizations:
- From 24: \( 2^3 \times 3^1 \)
- From 60: \( 2^2 \times 3^1 \)
- The common prime factors are:
- For 2: the minimum power is \( 2^2 \)
- For 3: the minimum power is \( 3^1 \)

**Step 4: Calculate the HCF**
- Multiply the common prime factors:
- HCF = \( 2^2 \times 3^1 = 4 \times 3 = 12 \)

### Part (b): Find the HCF of 48, 64, and 80

**Step 1: Prime Factorization of 48**
- Divide 48 by the smallest prime number, which is 2:
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 is a prime number, so we stop here.
- The prime factorization of 48 is:
- \( 48 = 2^4 \times 3^1 \)

**Step 2: Prime Factorization of 64**
- Divide 64 by the smallest prime number, which is 2:
- 64 ÷ 2 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1
- The prime factorization of 64 is:
- \( 64 = 2^6 \)

**Step 3: Prime Factorization of 80**
- Divide 80 by the smallest prime number, which is 2:
- 80 ÷ 2 = 40
- 40 ÷ 2 = 20
- 20 ÷ 2 = 10
- 10 ÷ 2 = 5
- 5 is a prime number, so we stop here.
- The prime factorization of 80 is:
- \( 80 = 2^4 \times 5^1 \)

**Step 4: Identify Common Factors**
- Now, we identify the common prime factors from the factorizations:
- From 48: \( 2^4 \times 3^1 \)
- From 64: \( 2^6 \)
- From 80: \( 2^4 \times 5^1 \)
- The common prime factor is:
- For 2: the minimum power is \( 2^4 \)

**Step 5: Calculate the HCF**
- HCF = \( 2^4 = 16 \)

### Final Answers
- (a) The HCF of 24 and 60 is **12**.
- (b) The HCF of 48, 64, and 80 is **16**.