What is the LCM for 4, 9, 12?

Multiples of #4# are #{4,8,12,16,20,24,28,32,color(red)36,40,44,48,52,56,60,64,68,72...}#

Multiples of #9# are #{9,18,27,color(red)36,45,54,63,72,81,90....}#

Multiples of #12# are #{12,24,color(red)36,48,60,72,84,96,108,120,....}#

Hence common multiples are #{color(red)36,72,----}#

and Least Common Multiple is #color(red)36#

Write each number as the product of its prime factors, then you know what you are working with.

Notice that you do not even need to consider #4#, because #4# is a factor of #12#, so any multiple of #12# will be a multiple of #4# as well.

#color(white)(............) 4 = 2xx2#
#color(white)(............) 9 =color(white)(xxxx.x)3xx3#
#color(white)(..........) 12 = 2xx2xx3#
#color(white)(.................) darrcolor(white)(.)darrcolor(white)(m)darrcolor(white)(.)darr#

#LCM =" " 2xx2xx3xx3 = 36#

Notice that in factor form:

#2xx2# is there for the #4#
#2xx2xx3# is there for the #12#
#3xx3# is there for the #9#

All the numbers are in the LCM, but there are no unnecessary factors.

Start with the largest figure.
In this case, that is 12.

You know that the LCM has to be some multiple of 12, so start considering the multiples of 12 one at a time.

#12 xx 1# #larr# 12 doesn't work because 9 doesn't go into 12 evenly (even though 4 does.)

#12 xx 2# #larr# 24 isn't it either because 9 doesn't go into 24

#12 xx 3# #larr# Here is the right answer.
4, 9, and 12 all go into 36 evenly
............................

Here's another example
Find the LCM of 4, 6, and 10

Don't waste your time fooling around with prime factors.
Instead, just rapidly consider each multiple of 10 until you hit on the one that works

10 --  thinking -- "Won't divide by 4 or 6"
20 -- thinking -- "Not divisible by 6"
30 -- thinking -- "Won't take 4"
40 -- thinking -- "Not divisible by 6"
50 ---thinking -- "Not divisible by 4 or 6"
60 -- #color(red)(Perfect!#

This usually takes only a few seconds, and you should always do it this way when the numbers let you.

If the numbers are too hard to work with like this, you can always just go back to finding prime factors again.

But the first choice of the way to find LCMs and LCDs should be just thinking in turn about each multiple of the largest figure.