A common task in math is to find the least common
multiple (or LCM as abbreviated by
Abbreviation Finder) of two numbers. If we do this for the
sake of adding or subtracting fractions with unlike
denominators (discussed in another article), then
this number will be used as the lowest common
denominator (or LCD as abbreviated by
Abbreviation Finder) for the two factions.
Note that there are many clever methods to calculate
the LCM of two numbers. Most of these are not at all
practical, since they take a very long time, and the
risk of error. They are typically only studied once,
when the subject will be first be taught, and then
never again. In later math and on standardized
exams, we are never given large and obscure figures
to work with. It is never the point. This article
shows you the steps to quickly find the LCM of two
rational numbers, maybe even in your head.
1. Let us find the LCM of 4 and 6. The first thing
to understand is that the LCM is not the same as the
greatest common factor (or GCF as abbreviated by
Abbreviation Finder). Students often get
this mixed up. The GCF is the largest number that
divides evenly into both numbers. In this case it is
2. To find the LCM, we must examine multiples of
each number, and see then find the lowest multiple
that each number has in common. When you are working
with GCF, think "less" and when working with LCM,
think "bigger".
2. Let's start by listing a few multiples of each
number. Later you will be able to do this in your
head. Multiples of 6 are 6 (even), 12, 18, 24, 30,
etc. multiples of 4 are 4 (even), 8, 12, 16, 20, 24,
28, 32, etc. Look to see what multiples appear on
both lists. We see 12 and 24 years. If we kept both
lists going, there would be infinitely more. For
example 36 is also a common multiple.
3. To find the LCM, we simply need the least common
multiple. In this case, it's 12. If we wanted to add
fractions with denominators of 4 and 6, we would
have to convert each fraction has a denominator of
12, which we would call the lowest common
denominator (LCD).
4. It is important to understand that although the
work of LCM is usually best, can we also typically
work with any common multiple, although it will
require additional steps. Also understand that if we
are struggling to find the LCM of two numbers, we
can always just multiply the two numbers together to
get some common multiple. It will usually not be the
smallest, but it will be a common multiple that we
can work with. For this example, it would be 24.
5. Let's try another example. What is the LCM of 5
and 10? Multiples of 5 are 5, 10, 15, 20, 25, 30,
etc. multiples of 10 are 10, 20, 30, 40, etc. The
LCM is actually 10, which happened to be one of the
original figures. Keep in mind that a number is
always a multiple of itself. Now don't get confused.
If we were asked the GCF of these two numbers, it
would be 5. Make sure you understand why.
6. Yet another example. What is the LCM of 7 and 11?
Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56,
63, 70, 77, 84, etc. are multiples of 11 are 11, 22,
33, 44, 55, 66, 77, 88, etc. The LCM is 77. It took
a lot of work. There are two shortcuts. Both of
these numbers is Prime, which is discussed in
another article. The LCM of two primes will always
be the product of the two numbers. A simpler rule to
follow is that if you have trouble finding the LCM
of two numbers, just multiply them, as described in
step 4. . The result may or may not be the LCM, but
it will be some common multiple that we can work
with. For example, if I was asked to add 1/13 +
1/14, I wouldn't bother wasting time on finding the
actual LCM. I would multiply 13 times 14 to get 182,
and I want to use it as my LCD. There may have been
a lower, but it will work just fine. We can always
reduce the fraction later if necessary (discussed in
another article).
7. Students should ensure that they are comfortable
with this topic, and that they know the difference
between LCM and GCF. This topic will come back again
when we put together and subtract fractions, and
certainly in algebra in a more abstract way, so
learn it now, while you're still working with simple
numbers out of context. |
