Find the prime factorisation of the following numbers:
(i) 68 `" "` (ii) 1248 `" "` (iii) 2304

Answer

To find the prime factorization of the numbers 68, 1248, and 2304, we will follow a systematic approach by dividing each number by the smallest prime number possible until we reach 1. Let's go through each number step by step.

### (i) Prime Factorization of 68

1. **Divide by 2 (smallest prime number)**:
- 68 ÷ 2 = 34
- Write down: 2

2. **Divide 34 by 2**:
- 34 ÷ 2 = 17
- Write down: 2

3. **Now we have 17, which is a prime number**:
- Write down: 17

4. **Combine the factors**:
- The prime factorization of 68 is: \( 2 \times 2 \times 17 \) or \( 2^2 \times 17 \).

### (ii) Prime Factorization of 1248

1. **Divide by 2**:
- 1248 ÷ 2 = 624
- Write down: 2

2. **Divide 624 by 2**:
- 624 ÷ 2 = 312
- Write down: 2

3. **Divide 312 by 2**:
- 312 ÷ 2 = 156
- Write down: 2

4. **Divide 156 by 2**:
- 156 ÷ 2 = 78
- Write down: 2

5. **Divide 78 by 2**:
- 78 ÷ 2 = 39
- Write down: 2

6. **Now divide 39 by 3 (next smallest prime)**:
- 39 ÷ 3 = 13
- Write down: 3

7. **Now we have 13, which is a prime number**:
- Write down: 13

8. **Combine the factors**:
- The prime factorization of 1248 is: \( 2^5 \times 3 \times 13 \).

### (iii) Prime Factorization of 2304

1. **Divide by 2**:
- 2304 ÷ 2 = 1152
- Write down: 2

2. **Divide 1152 by 2**:
- 1152 ÷ 2 = 576
- Write down: 2

3. **Divide 576 by 2**:
- 576 ÷ 2 = 288
- Write down: 2

4. **Divide 288 by 2**:
- 288 ÷ 2 = 144
- Write down: 2

5. **Divide 144 by 2**:
- 144 ÷ 2 = 72
- Write down: 2

6. **Divide 72 by 2**:
- 72 ÷ 2 = 36
- Write down: 2

7. **Divide 36 by 2**:
- 36 ÷ 2 = 18
- Write down: 2

8. **Divide 18 by 2**:
- 18 ÷ 2 = 9
- Write down: 2

9. **Now divide 9 by 3**:
- 9 ÷ 3 = 3
- Write down: 3

10. **Now we have 3, which is a prime number**:
- Write down: 3

11. **Combine the factors**:
- The prime factorization of 2304 is: \( 2^8 \times 3^2 \).

### Summary of Prime Factorizations:
- \( 68 = 2^2 \times 17 \)
- \( 1248 = 2^5 \times 3 \times 13 \)
- \( 2304 = 2^8 \times 3^2 \)