Find the L.C.M. and H.C.F. of 9, 117 and 729 by the prime factorisation method.

Answer

To find the L.C.M. (Least Common Multiple) and H.C.F. (Highest Common Factor) of the numbers 9, 117, and 729 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of the Given Numbers

1. **Prime Factorization of 9:**
- 9 can be expressed as \(3 \times 3\) or \(3^2\).

2. **Prime Factorization of 117:**
- 117 can be divided by 3:
- \(117 \div 3 = 39\)
- Now, factor 39:
- \(39 \div 3 = 13\)
- So, the prime factorization of 117 is \(3^2 \times 13\).

3. **Prime Factorization of 729:**
- 729 can be divided by 3 repeatedly:
- \(729 \div 3 = 243\)
- \(243 \div 3 = 81\)
- \(81 \div 3 = 27\)
- \(27 \div 3 = 9\)
- \(9 \div 3 = 3\)
- \(3 \div 3 = 1\)
- Therefore, \(729 = 3^6\).

Now we have the prime factorizations:
- \(9 = 3^2\)
- \(117 = 3^2 \times 13\)
- \(729 = 3^6\)

### Step 2: Finding H.C.F.

To find the H.C.F., we take the product of the lowest powers of the common prime factors.

- The common prime factor is \(3\).
- The lowest power of \(3\) in the factorizations is \(3^2\).

Thus, the H.C.F. is:
\[ \text{H.C.F.} = 3^2 = 9 \]

### Step 3: Finding L.C.M.

To find the L.C.M., we take the product of the highest powers of all prime factors present in the factorizations.

- For \(3\), the highest power is \(3^6\).
- For \(13\), the highest power is \(13^1\).

Thus, the L.C.M. is:
\[ \text{L.C.M.} = 3^6 \times 13^1 \]

Calculating this:
\[ 3^6 = 729 \]
\[ \text{L.C.M.} = 729 \times 13 = 9477 \]

### Final Results

- H.C.F. = 9
- L.C.M. = 9477