Find the LCM of 45 and 75 using division

Answer

To find the LCM (Least Common Multiple) of 45 and 75 using the division method, follow these steps:

### Step 1: Write the numbers
We start with the two numbers we need to find the LCM for:
- 45
- 75

### Step 2: Divide by the smallest prime number
We begin by dividing both numbers by the smallest prime number that can divide at least one of them. The smallest prime number is 3.
- 45 ÷ 3 = 15
- 75 ÷ 3 = 25

So, we write:
```
3 | 45 75
| 15 25
```

### Step 3: Continue dividing
Next, we check if we can divide the resulting numbers (15 and 25) by 3 again:
- 15 ÷ 3 = 5 (we can divide by 3)
- 25 ÷ 3 = 25 (we cannot divide by 3, so we keep it as is)

Now, we write:
```
3 | 45 75
| 15 25
3 | 5 25
```

### Step 4: Divide by the next smallest prime number
Now, we can divide by the next smallest prime number, which is 5:
- 5 ÷ 5 = 1
- 25 ÷ 5 = 5

Now, we write:
```
3 | 45 75
| 15 25
3 | 5 25
5 | 1 5
```

### Step 5: Continue dividing
Now, we can divide the next number (5) by 5:
- 5 ÷ 5 = 1
- 1 remains as 1.

Now, we write:
```
3 | 45 75
| 15 25
3 | 5 25
5 | 1 5
5 | 1 1
```

### Step 6: Stop when all numbers are 1
We stop when all the numbers in the last row are 1.

### Step 7: Multiply all the divisors
Now we multiply all the prime factors we used to divide:
- LCM = 3 × 3 × 5 × 5

### Step 8: Calculate the LCM
Now we calculate:
- 3 × 3 = 9
- 5 × 5 = 25
- 9 × 25 = 225

Thus, the LCM of 45 and 75 is **225**.