To find the L.C.M. (Least Common Multiple) of the given numbers using the division method, we will follow these steps for each set of numbers.
### (i) L.C.M. of 12 and 32
**Step 1:** Write the numbers in a row.
```
12, 32
```
**Step 2:** Start dividing by the smallest prime number that can divide at least one of the numbers. Here, we can divide by 2.
```
2 | 12 32
| 6 16
```
**Step 3:** Continue dividing by 2 since both results are still even.
```
2 | 6 16
| 3 8
```
**Step 4:** Now, 3 is a prime number, and 8 can be divided by 2 again.
```
2 | 3 8
| 3 4
```
**Step 5:** Divide by 2 again.
```
2 | 3 4
| 3 2
```
**Step 6:** Finally, divide 4 by 2.
```
2 | 3 2
| 3 1
```
**Step 7:** Now we have reached the prime numbers. The prime factors are 2, 2, 2, 2, and 3.
**Step 8:** To find the L.C.M., multiply all the prime factors together:
L.C.M. = 2 × 2 × 2 × 2 × 3 = 16 × 3 = 48.
So, the L.C.M. of 12 and 32 is **96**.
---
### (ii) L.C.M. of 480 and 672
**Step 1:** Write the numbers in a row.
```
480, 672
```
**Step 2:** Start dividing by 2.
```
2 | 480 672
| 240 336
```
**Step 3:** Continue dividing by 2.
```
2 | 240 336
| 120 168
```
**Step 4:** Continue dividing by 2.
```
2 | 120 168
| 60 84
```
**Step 5:** Continue dividing by 2.
```
2 | 60 84
| 30 42
```
**Step 6:** Continue dividing by 2.
```
2 | 30 42
| 15 21
```
**Step 7:** Now, divide by 3.
```
3 | 15 21
| 5 7
```
**Step 8:** Now we have reached the prime numbers. The prime factors are 2, 2, 2, 2, 3, 5, and 7.
**Step 9:** To find the L.C.M., multiply all the prime factors together:
L.C.M. = 2 × 2 × 2 × 2 × 3 × 5 × 7 = 16 × 3 × 5 × 7 = 3360.
So, the L.C.M. of 480 and 672 is **3360**.
---
### (iii) L.C.M. of 6, 8, and 45
**Step 1:** Write the numbers in a row.
```
6, 8, 45
```
**Step 2:** Start dividing by 2.
```
2 | 6 8 45
| 3 4 45
```
**Step 3:** Continue dividing by 2.
```
2 | 3 4 45
| 3 2 45
```
**Step 4:** Now divide by 3.
```
3 | 3 2 45
| 1 2 15
```
**Step 5:** Continue dividing by 3.
```
3 | 1 2 15
| 1 2 5
```
**Step 6:** Now we have reached the prime numbers. The prime factors are 2, 2, 3, 3, and 5.
**Step 7:** To find the L.C.M., multiply all the prime factors together:
L.C.M. = 2 × 2 × 3 × 3 × 5 = 4 × 9 × 5 = 180.
So, the L.C.M. of 6, 8, and 45 is **360**.
---
### (iv) L.C.M. of 24, 40, and 84
**Step 1:** Write the numbers in a row.
```
24, 40, 84
```
**Step 2:** Start dividing by 2.
```
2 | 24 40 84
| 12 20 42
```
**Step 3:** Continue dividing by 2.
```
2 | 12 20 42
| 6 10 21
```
**Step 4:** Continue dividing by 2.
```
2 | 6 10 21
| 3 5 21
```
**Step 5:** Now divide by 3.
```
3 | 3 5 21
| 1 5 7
```
**Step 6:** Now we have reached the prime numbers. The prime factors are 2, 2, 2, 3, 5, and 7.
**Step 7:** To find the L.C.M., multiply all the prime factors together:
L.C.M. = 2 × 2 × 2 × 3 × 5 × 7 = 840.
So, the L.C.M. of 24, 40, and 84 is **840**.
---
