What's the Lowest Common Multiple (LCM) of 12 and 15?

When we're looking at the LCM (Least Common Multiple), we're looking for a number that both 12 and 15 are a factor of. Oftentimes people simply assume that if we multiply the two together, we'll find it. In this case, it'd be #12xx15=180#. 180 is a multiple of both, but is it the least one? Let's look.

I start with a prime factorization of both numbers:

#12=2xx2xx3#

#15=3xx5#

To find the LCM, we want to have all the prime factors from both numbers accounted for.

For instance, there are two 2s (in the 12). Let's put those in:

#LCM=2xx2xx...#

There is one 3 in both the 12 and the 15, so we need one 3:

#LCM=2xx2xx3xx...#

And there is one 5 (in the 15) so let's put that in:

#LCM=2xx2xx3xx5=60#

#12xx5=60#
#15xx3=60#

another approach is to use teh relation

#ab=hcf(a,b)lcm(ab)#

now #hcf(12,15)=3#

#:.12xx15=3xxlcm(12,15)#

#lcm(12,15)=(cancel(12)^4xx15)/cancel(3)#

#lcm=4xx15=60#

The LCM is the least common multiple. We can find the LCM by listing the multiples of the two numbers and identifying the lowest multiple they have in common.

#12:##12,24,36,48,color(red)60,72,84...#

#15:##15,30,45,color(red)60...#

The LCM is #60#.

Let's try to find the LCM of #12# and #15#.

We get:
#12=2*2*color(blue)3#
#15= color(blue)3*5#

We see that they both share #3# is their LCM.

We divide each number by their LCM.

#12/3=>4#

#15/3=>5#

We multiply these two quotients and the LCM to get our final answer:

#3*4*5=60#

That is our answer!