Find the HCF of the following numbers using:
(a) prime factorisation (b) division method
144, 234

Answer

To find the HCF (Highest Common Factor) of the numbers 144 and 234 using two methods: prime factorization and the division method, we can follow these steps:

### (a) Prime Factorization Method

1. **Prime Factorization of 144**:
- Start dividing 144 by the smallest prime number, which is 2.
- \( 144 \div 2 = 72 \)
- \( 72 \div 2 = 36 \)
- \( 36 \div 2 = 18 \)
- \( 18 \div 2 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- Thus, the prime factorization of 144 is:
\[ 144 = 2^4 \times 3^2 \]

2. **Prime Factorization of 234**:
- Start dividing 234 by the smallest prime number, which is 2.
- \( 234 \div 2 = 117 \)
- \( 117 \div 3 = 39 \)
- \( 39 \div 3 = 13 \)
- \( 13 \div 13 = 1 \)
- Thus, the prime factorization of 234 is:
\[ 234 = 2^1 \times 3^2 \times 13^1 \]

3. **Finding Common Factors**:
- Now, we identify the common prime factors between 144 and 234.
- The common prime factors are:
- \( 2^1 \) (the minimum power of 2)
- \( 3^2 \) (the minimum power of 3)
- Therefore, the HCF is:
\[ HCF = 2^1 \times 3^2 = 2 \times 9 = 18 \]

### (b) Division Method

1. **Set Up the Division**:
- We will divide the larger number (234) by the smaller number (144).
- \( 234 \div 144 \) gives a quotient of 1 and a remainder of:
\[ 234 - 144 = 90 \]

2. **Continue Dividing**:
- Now, take the previous divisor (144) and divide it by the remainder (90).
- \( 144 \div 90 \) gives a quotient of 1 and a remainder of:
\[ 144 - 90 = 54 \]

3. **Repeat the Process**:
- Now, divide 90 by 54.
- \( 90 \div 54 \) gives a quotient of 1 and a remainder of:
\[ 90 - 54 = 36 \]
- Next, divide 54 by 36.
- \( 54 \div 36 \) gives a quotient of 1 and a remainder of:
\[ 54 - 36 = 18 \]
- Finally, divide 36 by 18.
- \( 36 \div 18 \) gives a quotient of 2 and a remainder of:
\[ 36 - 36 = 0 \]

4. **Conclusion**:
- Since the last non-zero remainder is 18, the HCF of 144 and 234 is:
\[ HCF = 18 \]

### Final Answer
The HCF of 144 and 234 is **18**.