Least Common Multiple
(from mathsisfun.com)
The smallest (non-zero) number that is a multiple of two or more numbers.
Least Common Multiple is made up of the words Least, Common and Multiple:
What is a "Multiple" ?
The multiples of a number are what you get when you multiply it by other
numbers (such as if you multiply it by 1,2,3,4,5, etc). Just like the multiplication
table.
Here are some examples:
What is a "Common Multiple" ?
When you list the multiples of two (or more) numbers, and find the same value
in both lists, then that is acommon multiple of those numbers.
For example, when you write down the multiples of 4 and 5,
the common multiples are those that are found in both lists:
Notice that 20 and 40 appear in both lists?
So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too)
What is the "Least Common Multiple" ?
It is simply the smallest of the common multiples.
In our previous example, the smallest of the common multiples is 20 ...
... so the Least Common Multiple of 4 and 5 is 20.
Finding the Least Common Multiple
It is a really easy thing to do. Just start listing the multiples of the numbers until
you get a match.
Example: Find the least common multiple for 3 and 5:
The multiples of 3 are 3, 6, 9, 12, 15, ...,
and the multiples of 5 are 5, 10, 15, 20, ..., like this:
As you can see on this number line, the first time the multiples match up is
15. Answer: 15
More than 2 Numbers
Hint: You can have smaller lists for the bigger numbers.
Why is this Useful?
One of the most useful things is when we want to find the common denominator:
Example: Compare
!
!
to
!
!
?
Comparing is MUCH EASIER when the fractions are rewritten using equivalent
fractions so that the 2 have common denominators.
In this case, the denominators are 5 and 3. The lowest common denominator is
simply the Lowest common multiple of 5 and 3. When is 15. So we can make
each fraction over 15.
!
!
!
!
becomes
becomes
!
!"
!"
!"
Now it becomes obvious that 2/3 is greater, as it is worth 10 fifteenths compared
to 3/5 which is only worth 9 fifteenths.
Using the LCM in worded questions
Whenever we have 2 cycles of different lengths and we want to know when the 2
cycles will finish at the same time, the LCM can be used.
Example: if 2 swimmers take 40 seconds, and 50 seconds respectively to
do a lap, then…
Lap Times for Swimmer 1 will be: 40, 80, 120, 160, 200 (multiples of 40).
Lap Times for Swimmer 2 will be: 50, 100, 150, 200 (multiples of 50).
NOTE: The Lowest Common Multiple, 200 in this case, tells me after how
many seconds they will finish a lap together. And its after 200 seconds (or
3 minutes and 20 seconds).
* Extra: Using the Highest Common Factor to find the LCM
There is another method for finding the LCM than simply comparing lists: you can
use the HCF.
The rule is this:
!"# !" ‘!’ !"# ‘!’ = ! × !
!"# !" ‘!’ !"# ‘!’
This says, multiple the 2 numbers together, and then divide by the highest
common factor.
CAUTION:
Many students mistakenly just do the top part… multiply the 2 numbers together.
And occasionally, it works.
Example: Find the LCM of 2 and 7. 2 x 7 = 14. And indeed the LCM of
2 and 7 is 14.
HOWEVER, it only worked because 2 and 7 had no common factors except 1.
In the following example, it won’t work to just multiply.
Example: Find the LCM of 8 and 12. 8 x 12 = 96. But the LCM is 24.
THINK: 96 is a common multiple, BUT… since 4 is a factor of both, if we
divide 96 by 4, the result will be a common factor of 8 and 12.
So 96 divided by 4 = 24. And indeed this is the answer.
DON’T FORGET TO DIVIDE BY THE HCF when using this method
* Extra: Using Scratch to find LCM
Using our HCF program from before (See Highest Common Factor notes if you
haven’t made it yet), once we find the HCF, the LCM is just the product of the 2
numbers, divided by the HCF.
So if we open up our program we get this:
Now, instea of saying the HCF in the last line, we can make a variable called HCF
to hold onto the value of the HCF.
The only problem is, we have LOST our original 2 numbers because we kept
changing a and b. We will need to make 2 more variables to store them (1st
number, 2nd number) and store them at the start straight after the user gives the
numbers.
Note that after the user gives the number, we store it twice. Once to HOLD
ONTO, and the other to play with as we find the HCF.
OK, so now we have the 2 numbers AND the HCF.
LCM = 1st number x 2nd number divided by HCF. So this should be easy.
To do order of operations, we put operations INSIDE other operations brackets.
e.g.
Then to divide THE RESULT by the hcf. We do this by getting the division block
and making the denominator ‘hcf’.
We drag the
The final code is this:
block to the numerator to get this: