Using prime factorisation method, find the HCF and LCM of 210 and 175

Answer

To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of 210 and 175 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of 210
1. Divide 210 by the smallest prime number, which is 2:
\[ 210 \div 2 = 105 \]
2. Next, divide 105 by the smallest prime number that divides it, which is 3:
\[ 105 \div 3 = 35 \]
3. Now, divide 35 by the next smallest prime number, which is 5:
\[ 35 \div 5 = 7 \]
4. Finally, divide 7 by itself:
\[ 7 \div 7 = 1 \]
5. So, the prime factorization of 210 is:
\[ 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 \]

### Step 2: Prime Factorization of 175
1. Start by dividing 175 by the smallest prime number, which is 5:
\[ 175 \div 5 = 35 \]
2. Next, divide 35 by 5 again:
\[ 35 \div 5 = 7 \]
3. Finally, divide 7 by itself:
\[ 7 \div 7 = 1 \]
4. So, the prime factorization of 175 is:
\[ 175 = 5^2 \times 7^1 \]

### Step 3: Finding the HCF
1. Identify the common prime factors from the factorizations:
- From 210: \(2^1, 3^1, 5^1, 7^1\)
- From 175: \(5^2, 7^1\)
2. The common prime factors are 5 and 7.
3. Take the lowest power of each common prime factor:
- For 5: \(5^1\)
- For 7: \(7^1\)
4. Multiply these together to find the HCF:
\[ \text{HCF} = 5^1 \times 7^1 = 5 \times 7 = 35 \]

### Step 4: Finding the LCM
1. Take all the prime factors from both numbers, using the highest power of each:
- From 210: \(2^1, 3^1, 5^1, 7^1\)
- From 175: \(5^2, 7^1\)
2. The highest powers are:
- \(2^1\)
- \(3^1\)
- \(5^2\)
- \(7^1\)
3. Multiply these together to find the LCM:
\[ \text{LCM} = 2^1 \times 3^1 \times 5^2 \times 7^1 = 2 \times 3 \times 25 \times 7 \]
4. Calculate it step by step:
- \(2 \times 3 = 6\)
- \(6 \times 25 = 150\)
- \(150 \times 7 = 1050\)
5. Thus, the LCM is:
\[ \text{LCM} = 1050 \]

### Final Answer
- HCF of 210 and 175 is **35**.
- LCM of 210 and 175 is **1050**.