Find the H.C.F. of 140, 210 and 350 by prime factorisation method.

Answer

To find the H.C.F. (Highest Common Factor) of 140, 210, and 350 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of 140
- Start by dividing 140 by the smallest prime number, which is 2.
- \( 140 \div 2 = 70 \)
- Divide 70 by 2 again.
- \( 70 \div 2 = 35 \)
- Now, 35 is not divisible by 2, so we move to the next prime number, which is 3. It is also not divisible by 3, so we try 5.
- \( 35 \div 5 = 7 \)
- Finally, 7 is a prime number.
- Thus, the prime factorization of 140 is:
\[ 140 = 2^2 \times 5^1 \times 7^1 \]

### Step 2: Prime Factorization of 210
- Start by dividing 210 by 2.
- \( 210 \div 2 = 105 \)
- Next, 105 is not divisible by 2, so we try 3.
- \( 105 \div 3 = 35 \)
- Now, divide 35 by 5.
- \( 35 \div 5 = 7 \)
- Finally, 7 is a prime number.
- Thus, the prime factorization of 210 is:
\[ 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 \]

### Step 3: Prime Factorization of 350
- Start by dividing 350 by 2.
- \( 350 \div 2 = 175 \)
- Now, 175 is not divisible by 2, so we try 3, which it is not divisible by either. Next, we try 5.
- \( 175 \div 5 = 35 \)
- Now, divide 35 by 5 again.
- \( 35 \div 5 = 7 \)
- Finally, 7 is a prime number.
- Thus, the prime factorization of 350 is:
\[ 350 = 2^1 \times 5^2 \times 7^1 \]

### Step 4: Identify Common Prime Factors
Now we have the prime factorizations:
- \( 140 = 2^2 \times 5^1 \times 7^1 \)
- \( 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 \)
- \( 350 = 2^1 \times 5^2 \times 7^1 \)

### Step 5: Find the H.C.F.
To find the H.C.F., we take the lowest power of each common prime factor:
- For 2: the lowest power is \( 2^1 \)
- For 5: the lowest power is \( 5^1 \)
- For 7: the lowest power is \( 7^1 \)

Thus, the H.C.F. is:
\[ H.C.F. = 2^1 \times 5^1 \times 7^1 = 2 \times 5 \times 7 = 70 \]

### Final Answer
The H.C.F. of 140, 210, and 350 is **70**.

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