Find the G.C.D. of the given numbers by prime factorisation method:
(i) 24, 45 `" "` (ii) 180, 252, 324

Answer

To find the G.C.D. (Greatest Common Divisor) of the given numbers using the prime factorization method, we will follow these steps:

### (i) Finding G.C.D. of 24 and 45

**Step 1: Prime Factorization of 24**
- 24 can be divided by 2:
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 is a prime number.
- Therefore, the prime factorization of 24 is:
\[ 24 = 2^3 \times 3^1 \]

**Step 2: Prime Factorization of 45**
- 45 can be divided by 3:
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 is a prime number.
- Therefore, the prime factorization of 45 is:
\[ 45 = 3^2 \times 5^1 \]

**Step 3: Identify Common Factors**
- The prime factors of 24 are \(2^3\) and \(3^1\).
- The prime factors of 45 are \(3^2\) and \(5^1\).
- The common prime factor is 3.

**Step 4: Find the G.C.D.**
- The G.C.D. is obtained by taking the lowest power of the common prime factors:
\[ G.C.D. = 3^1 = 3 \]

### (ii) Finding G.C.D. of 180, 252, and 324

**Step 1: Prime Factorization of 180**
- 180 can be divided by 2:
- 180 ÷ 2 = 90
- 90 ÷ 2 = 45
- 45 can be divided by 3:
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- Therefore, the prime factorization of 180 is:
\[ 180 = 2^2 \times 3^2 \times 5^1 \]

**Step 2: Prime Factorization of 252**
- 252 can be divided by 2:
- 252 ÷ 2 = 126
- 126 ÷ 2 = 63
- 63 can be divided by 3:
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- Therefore, the prime factorization of 252 is:
\[ 252 = 2^2 \times 3^2 \times 7^1 \]

**Step 3: Prime Factorization of 324**
- 324 can be divided by 2:
- 324 ÷ 2 = 162
- 162 ÷ 2 = 81
- 81 can be divided by 3:
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- Therefore, the prime factorization of 324 is:
\[ 324 = 2^2 \times 3^4 \]

**Step 4: Identify Common Factors**
- The prime factors of 180 are \(2^2\), \(3^2\), and \(5^1\).
- The prime factors of 252 are \(2^2\), \(3^2\), and \(7^1\).
- The prime factors of 324 are \(2^2\) and \(3^4\).
- The common prime factors are \(2^2\) and \(3^2\).

**Step 5: Find the G.C.D.**
- The G.C.D. is obtained by taking the lowest power of the common prime factors:
\[ G.C.D. = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Final Answers:
- G.C.D. of 24 and 45 is **3**.
- G.C.D. of 180, 252, and 324 is **36**.