Certainly! Let's find the Least Common Multiple (LCM) of 20, 24, and 45 using the division method. This method involves dividing the given numbers by their common prime factors until we are left with 1 in the quotient.
1. Write down the numbers to be divided:
- 20, 24, 45
2. Find the smallest prime number that can divide at least one of the given numbers:
- The smallest prime number is 2.
3. Divide the numbers by 2 (if possible):
- [tex]\(20 \div 2 = 10\)[/tex]
- [tex]\(24 \div 2 = 12\)[/tex]
- [tex]\(45\)[/tex] (cannot be divided by 2, so it remains as it is)
The step looks like this:
```
2 | 20 24 45
--------------
| 10 12 45
```
4. Continue dividing the resulting numbers by the smallest prime number:
- Again, we use 2 since it is still a factor for some of the numbers:
- [tex]\(10 \div 2 = 5\)[/tex]
- [tex]\(12 \div 2 = 6\)[/tex]
- [tex]\(45\)[/tex] (remains as it is)
The step looks like this:
```
2 | 10 12 45
--------------
| 5 6 45
```
5. Repeat the process with the smallest prime number that can divide at least one of the remaining numbers:
- We continue with 2:
- [tex]\(6 \div 2 = 3\)[/tex], [tex]\(5\)[/tex] and [tex]\(45\)[/tex] remain as they are.
The step looks like this:
```
2 | 5 6 45
--------------
| 5 3 45
```
6. Now switch to the next smallest prime number, which is 3:
- [tex]\(3 \div 3 = 1\)[/tex], [tex]\(5\)[/tex] and [tex]\(45 \div 3 = 15\)[/tex]
The step looks like this:
```
3 | 5 3 45
--------------
| 5 1 15
```
7. Continue with the prime number 3:
- [tex]\(15 \div 3 = 5\)[/tex], [tex]\(5\)[/tex] and [tex]\(1\)[/tex] remain as they are.
The step looks like this:
```
3 | 5 1 15
--------------
| 5 1 5
```
8. Now switch to the prime number 5:
- [tex]\(5 \div 5 = 1\)[/tex]
The step looks like this:
```
5 | 5 1 5
--------------
| 1 1 1
```
9. Multiply all the prime factors used to divide the numbers to get the LCM:
- The prime factors we used were 2, 2, 2, 3, 3, and 5.
- [tex]\(LCM = 2 \times 2 \times 2 \times 3 \times 3 \times 5\)[/tex]
10. Calculate the final result:
- [tex]\( 2 \times 2 = 4\)[/tex]
- [tex]\( 4 \times 2 = 8\)[/tex]
- [tex]\( 8 \times 3 = 24\)[/tex]
- [tex]\( 24 \times 3 = 72\)[/tex]
- [tex]\( 72 \times 5 = 360\)[/tex]
So, the LCM of 20, 24, and 45 is 360.
