Find the H.C.F. of (8,648) by prime factorisation and hence find the L.C.M.

Answer

To find the H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) of the numbers 8 and 648 using prime factorization, follow these steps:

### Step 1: Prime Factorization of 8
1. Start with the number 8.
2. Divide by the smallest prime number, which is 2.
- \( 8 \div 2 = 4 \)
3. Divide 4 by 2 again.
- \( 4 \div 2 = 2 \)
4. Divide 2 by 2 once more.
- \( 2 \div 2 = 1 \)

So, the prime factorization of 8 is:
\[ 8 = 2^3 \]

### Step 2: Prime Factorization of 648
1. Start with the number 648.
2. Divide by the smallest prime number, which is 2.
- \( 648 \div 2 = 324 \)
3. Divide 324 by 2 again.
- \( 324 \div 2 = 162 \)
4. Divide 162 by 2 once more.
- \( 162 \div 2 = 81 \)
5. Now, divide 81 by the next smallest prime number, which is 3.
- \( 81 \div 3 = 27 \)
6. Divide 27 by 3 again.
- \( 27 \div 3 = 9 \)
7. Divide 9 by 3 once more.
- \( 9 \div 3 = 3 \)
8. Finally, divide 3 by 3.
- \( 3 \div 3 = 1 \)

So, the prime factorization of 648 is:
\[ 648 = 2^3 \times 3^4 \]

### Step 3: Finding the H.C.F.
1. Identify the common prime factors from the factorizations of 8 and 648.
- The common prime factor is \( 2 \).
2. Take the lowest power of the common prime factor.
- For \( 2 \), the lowest power is \( 2^3 \).

Thus, the H.C.F. of 8 and 648 is:
\[ \text{H.C.F.} = 2^3 = 8 \]

### Step 4: Finding the L.C.M.
1. Use the relationship between H.C.F. and L.C.M.:
\[ \text{L.C.M.} \times \text{H.C.F.} = \text{Product of the Numbers} \]
2. Substitute the known values:
\[ \text{L.C.M.} \times 8 = 8 \times 648 \]
3. Calculate the product:
\[ 8 \times 648 = 5184 \]
4. Now, divide by the H.C.F. to find the L.C.M.:
\[ \text{L.C.M.} = \frac{5184}{8} = 648 \]

### Final Answer
- H.C.F. of 8 and 648 is **8**.
- L.C.M. of 8 and 648 is **648**.