Aaron, ruth and walter created shortcuts for finding the least common multiple of two numbers. do their methods work for any pair of numbers? explain why it works

L.C.M= Least common multiple

If we have to find least common multiple of two numbers  

you have to check out which kind of numbers they are.

1. If the numbers are co- prime= Multiply the two numbers  

you will get the L.C.M.[Co-Prime numbers→If two numbers doesn't have any common factor then,that two numbers are Co-prime.]

For example L.C.M

(2,3)=2×3=6

(4,5)=4×5=20

(7,9)=7×9=63 So , you can find many more examples.

2. If the two numbers are not Co-prime like (4,6),(8,10)

Then, there is another procedure.

First you have to factorize the two numbers.Then you have to take out common numbers and and then multiply the numbers which are not common with the common number.

For example ,

L.C.M (12,16),

12=2×2×3

16=2×2×2×2

Common numbers are 2×2.

Non common numbers are  3×2×2.

So, L.C.M=2×2×2×2×3=48

Take another example L.C.M(40,80)

40=2×2×2×5

80=2×2×2×2×5

Common numbers=2×2×2×5

Non common number=2

So, L.C.M=2×2×2×2×5=80

If  Aaron, Ruth and Walter created this  shortcut for finding the least common multiple of any two numbers,then definitely this will work for any pair of numbers.