Find the H.C.F. and L.C.M. of 30, 72 and 432 using the prime factorisation method.

Answer

To find the H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) of the numbers 30, 72, and 432 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of Each Number

**For 30:**
- The prime factors of 30 can be found by dividing by the smallest prime numbers.
- 30 = 2 × 15
- 15 = 3 × 5
- Therefore, the prime factorization of 30 is:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

**For 72:**
- Similarly, we factor 72.
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- Therefore, the prime factorization of 72 is:
\[ 72 = 2^3 \times 3^2 \]

**For 432:**
- Now, we factor 432.
- 432 = 2 × 216
- 216 = 2 × 108
- 108 = 2 × 54
- 54 = 2 × 27
- 27 = 3 × 9
- 9 = 3 × 3
- Therefore, the prime factorization of 432 is:
\[ 432 = 2^4 \times 3^3 \]

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

To find the H.C.F., we take the lowest power of each common prime factor from the factorizations.

- For the prime factor 2:
- The powers are \(2^1\) (from 30), \(2^3\) (from 72), and \(2^4\) (from 432).
- The minimum power is \(2^1\).

- For the prime factor 3:
- The powers are \(3^1\) (from 30), \(3^2\) (from 72), and \(3^3\) (from 432).
- The minimum power is \(3^1\).

- For the prime factor 5:
- The powers are \(5^1\) (from 30), \(5^0\) (from 72), and \(5^0\) (from 432).
- The minimum power is \(5^0\) (which means it is not included).

Thus, the H.C.F. is:
\[ H.C.F. = 2^1 \times 3^1 = 2 \times 3 = 6 \]

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

To find the L.C.M., we take the highest power of each prime factor from the factorizations.

- For the prime factor 2:
- The powers are \(2^1\) (from 30), \(2^3\) (from 72), and \(2^4\) (from 432).
- The maximum power is \(2^4\).

- For the prime factor 3:
- The powers are \(3^1\) (from 30), \(3^2\) (from 72), and \(3^3\) (from 432).
- The maximum power is \(3^3\).

- For the prime factor 5:
- The powers are \(5^1\) (from 30), \(5^0\) (from 72), and \(5^0\) (from 432).
- The maximum power is \(5^1\).

Thus, the L.C.M. is:
\[ L.C.M. = 2^4 \times 3^3 \times 5^1 \]
Calculating this:
- \(2^4 = 16\)
- \(3^3 = 27\)
- \(5^1 = 5\)

Now, multiplying these together:
\[ L.C.M. = 16 \times 27 \times 5 \]
Calculating \(16 \times 27 = 432\) and then \(432 \times 5 = 2160\).

### Final Answer:
- H.C.F. = 6
- L.C.M. = 2160