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Question

Find Least Common Multiple of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22 using prime factorisation method.

Answer

LCM( 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 ) = 232792560

Explanation

Step 1: Write down factorisation of each number:

2 = 2

3 = 3

4 = 2 · 2

5 = 5

6 = 2 · 3

7 = 7

8 = 2 · 2 · 2

9 = 3 · 3

10 = 2 · 5

11 = 11

12 = 2 · 2 · 3

13 = 13

14 = 2 · 7

15 = 3 · 5

16 = 2 · 2 · 2 · 2

17 = 17

18 = 2 · 3 · 3

19 = 19

20 = 2 · 2 · 5

21 = 3 · 7

22 = 2 · 11

Step 2 : Match primes vertically:

2=2                      
3=        3              
4=2·2                    
5=            5          
6=2·      3              
7=              7        
8=2·2·2                  
9=        3·3            
10=2·          5          
11=                11      
12=2·2·    3              
13=                  13    
14=2·            7        
15=        3·  5          
16=2·2·2·2                
17=                    17  
18=2·      3·3            
19=                      19
20=2·2·        5          
21=        3·    7        
22=2·              11      

Step 3 : Bring down numbers in each column and multiply to get LCM:

2=2                        
3=        3                
4=2·2                      
5=            5            
6=2·      3                
7=              7          
8=2·2·2                    
9=        3·3              
10=2·          5            
11=                11        
12=2·2·    3                
13=                  13      
14=2·            7          
15=        3·  5            
16=2·2·2·2                  
17=                    17    
18=2·      3·3              
19=                      19  
20=2·2·        5            
21=        3·    7          
22=2·              11        
LCM=2·2·2·2·3·3·5·7·11·13·17·19=232792560
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