Question

What is the least common multiple (LCM) of 8, 12, and 18?

Answer 1

Least Common Multiple is `72`

Explanation:

Multiples of `8` are `{8,16,24,32,40,48,56,64,72,80,88,96,104,112,120,128,136,144......}`

Multiples of `12` are `{12,24,36,48,60,72,84,96,108,120,132,144......}`

Multiples of `18` are `{18,36,54,72,90,108,126,144......}`

Hence common multiples are `{72,144,.......}`

and Least Common Multiple is `72`

Another shorter way is to write numbers as multiplication of its prime factors

`8=2xx2xx2` `->` `2` comes three times

`12=2xx2xx3` `->` `2` comes twice and `3` once.

`18=2xx3xx3` `->` `2` comes once and `3` twice

So maximum times is three times for `2` and two times for `3`.

Hence, Least Common Multiple is `2xx2xx2xx3xx3=72`.

Answer 2

Use Shwetyank's method in the class and an exam. The marking schema will be set up for that approach.

`color(blue)("My alternative approach - helps with understanding.")`

Explanation:

Target: LCM of 8,12 and 18.
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`color(magenta)("Once you are used to this it should look like")` `color(magenta)("what follows (for these numbers).")`

`12=color(white)("d")8+4 color(white)("d") ->2xx4=color(white)(..)8" so "2xx12=24`
`24=18+6 color(white)("d")->3xx6=color(white)(.)18" so "color(white)(.)3xx24=72`

`"LCM "=72`

3 lines inclusive of the answer for 3 relationships

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`color(blue)("Preamble about the method")`

There are many ways of writing any particular value so we can make it look however we wish as long as final outcome is the correct value. Example: 6 can and may be written as `5+1` if we so chose.

We can use this to our advantage.
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`color(blue)("Answering the question")`

`color(brown)("Dealing with the 8 and 12 part")`

Write 12 as `8+4`

Now we start counting the 12's but make sure it encompasses the 8 + 4's. When we have accumulated enough 4's to make another eight we have found the least common factor of 8 and 12

`1: -> color(green)(12)=color(red)(8)+4`
`2:-> ul(color(green)(12))=color(red)(8)+ul(4 larr" Add the 12's and the 4's")`
`color(white)("ddddd")color(green)(24) color(white)("ddd")ul(color(white)("ddd")color(red)(8))`
`color(white)("dddddddddddd")color(red)(24) larr" Adding all the 8's"`

`2" of "12=24`
`3" of "color(white)("d")8=24`

So LCF (for 8 and 12) is 24

So we have a fixed ratio of 8 and 12 at every sum of 24. Consequently any other product of factors will need to include some multiple of 24
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`color(brown)("Dealing with the 18 part")`

As the value of 24 locks together the counts of 8 and 12 we carry that forward.

Write the 24 as `18+6`

We are now counting the 24's

`1: ->color(green)( 24)=color(red)(18)+6`
`2: ->color(green)(24)=color(red)(18)+6`
`3:->ul(color(green)(24))=color(red)(18)+ul(6larr" Add the 24's and 6's")`
`color(white)("ddddd") color(green)(72)color(white)("dddddd")ul(color(red)(18))larr" Add all the 18's"`
`color(white)("ddddddddddddd")color(red)(72)`
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`color(magenta)(" LCM "= 72)`
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