Least Common Multiple

Least Common Multiple

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:
The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, etc ...
The multiples of 12 are: 12, 24, 36, 48, 60, 72, etc...

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 a common 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:
The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,...
The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...

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:
multiples

As you can see on this number line, the first time the multiples match up is 15. Answer: 15
More than 2 Numbers

You can also find the least common multiple of 3 (or more) numbers.
Example: Find the least common multiple for 4, 6, and 8

Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Multiples of 6 are: 6, 12, 18, 24, 30, 36, ...
Multiples of 8 are: 8, 16, 24, 32, 40, ....

So, 24 is the least common multiple (I can't find a smaller one !)

Hint: You can have smaller lists for the bigger numbers.
Least Common Multiple Tool

There is a another method, you can use our Least Common Multiple Tool to find it automatically.

Fractions

Fractions

A fraction is a part of a whole
Slice a pizza, and you will have fractions:

1/2 1/4 3/8
(One-Half)

(One-Quarter)

(Three-Eighths)

The top number tells how many slices you have
The bottom number tells how many slices the pizza was cut into.

Numerator / Denominator

We call the top number the Numerator, it is the number of parts you have.
We call the bottom number the Denominator, it is the number of parts the whole is divided into.
Numerator
Denominator

You just have to remember those names! (If you forget just think "Down"-ominator)
Equivalent Fractions

Some fractions may look different, but are really the same, for example:

4/8 = 2/4 = 1/2
(Four-Eighths) Two-Quarters) (One-Half)
= =

It is usually best to show an answer using the simplest fraction ( 1/2 in this case ). That is called Simplifying, or Reducing the Fraction
Adding Fractions

You can add fractions easily if the bottom number (the denominator) is the same:
1/4 + 1/4 = 2/4 = 1/2
(One-Quarter) (One-Quarter) (Two-Quarters) (One-Half)
+ = =

Another example:
5/8 + 1/8 = 6/8 = 3/4
+ = =

Adding Fractions with Different Denominators

But what if the denominators (the bottom numbers) are not the same? As in this example:

3/8 + 1/4 = ?
+ =

You must somehow make the denominators the same.

In this case it is easy, because we know that 1/4 is the same as 2/8 :

3/8 + 2/8 = 5/8
+ =

In that example it was easy to make the denominators the same, but it can be harder ... so you may need to use either of these methods:

* Least Common Denominator, or
* Common Denominator

to make them the same (they both work, use whichever you prefer).

Convert Decimals to Fractions

Convert Decimals to Fractions

(Multiply top and bottom by 10 until you get a whole number, then simplify)
To convert a Decimal to a Fraction follow these steps:
Step 1: Write down the decimal divided by 1, like this: decimal/1
Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there are two numbers after the decimal point, then use 100, if there are three then use 1000, etc.)
Step 3: Simplify (or reduce) the fraction


Example: Express 0.75 as a fraction

Step 1: Write down 0.75 divided by 1:
0.75
1

Step 2: Multiply both top and bottom by 100 (there were 2 digits after the decimal point so that is 10×10=100):
× 100
0.75 = 75

1 100
× 100

(Do you see how it turns the top number
into a whole number?)

Step 3: Simplify the fraction (this took me two steps):
÷5 ÷ 5

75 = 15 = 3
___ ___
100 20 4

÷5 ÷ 5


Answer = 3/4



Note: 75/100 is called a decimal fraction and 3/4 is called a common fraction !



Example: Express 0.625 as a fraction

Step 1: write down:
0.625
1

Step 2: multiply both top and bottom by 1,000 (there were 3 digits after the decimal point so that is 10×10×10=1,000)
625
1,000

Step 3: Simplify the fraction (it took me two steps here):
÷ 25 ÷ 5

625 = 25 = 5

1,000 40 8

÷ 25 ÷ 5


Answer = 5/8


Example: Express 0.333 as a fraction

Step 1: Write down:
0.333
1

Step 2: Multiply both top and bottom by 1,000 (there were 3 digits after the decimal point so that is 10×10×10=1,000)
333
1,000

Step 3: Simplify Fraction:

Can't get any simpler!


Answer = 333/1,000

But a Special Note:

If you really meant 0.333... (in other words 3s repeating forever which is called 3 recurring) then we need to follow a special argument. In this case we would write down:

0.333...
1

Then MULTIPLY both top and bottom by 3:
× 3
0.333... = 0.999...

1 3
× 3

And 0.999... = 1 (Does it? - see the 9 Recurring discussion for more if you are interested), so:

Answer = 1/3

Rounding Numbers

What is "Rounding" ?
Rounding means reducing the digits in a number while trying to keep its value similar. The result is less accurate, but easier to use.

Example: 73 rounded to the nearest ten is 70, because 73 is closer to 70 than to 80.


Common Method

There are several different methods for rounding, but here we will only look at the common method, the one used by most people ...
How to Round Numbers

* Decide which is the last digit to keep
* Leave it the same if the next digit is less than 5 (this is called rounding down)
* But increase it by 1 if the next digit is 5 or more (this is called rounding up)

Example: Round 74 to the nearest 10

* We want to keep the "7" as it is in the 10s position
* The next digit is "4" which is less than 5, so no change is needed to "7"

Answer: 70

(74 gets "rounded down")
Example: Round 86 to the nearest 10

* We want to keep the "8"
* The next digit is "6" which is 5 or more, so increase the "8" by 1 to "9"

Answer: 90

(86 gets "rounded up")

So: when the first digit removed is 5 or more, increase the last digit remaining by 1.
Why does 5 go up ?

5 is in the middle ... so we could go up or down. But we need a method that everyone can agree to always use.

So think about sport: you should have the same number of players on each team, right?


* 0,1,2,3 and 4 are on team "down"
* 5,6,7,8 and 9 are on team "up"

And that is the most important part of the "common" method of rounding. It is not a perfect method, but it is the one most people use.
Rounding Decimals
First you need to know if you are rounding to tenths, or hundredths, etc. Or maybe to "so many decimal places". That tells you how much of the number will be left when you finish.


Examples Because ...
3.1416 rounded to hundredths is 3.14 ... the next digit (1) is less than 5
1.2635 rounded to tenths is 1.3 ... the next digit (6) is 5 or more
1.2635 rounded to 3 decimal places is 1.264 ... the next digit (5) is 5 or more
Rounding Whole Numbers

You may want to round to tens, hundreds, etc, In this case you replace the removed digits with zero.
Examples Because ...
134.9 rounded to tens is 130 ... the next digit (4) is less than 5
12,690 rounded to thousands is 13,000 ... the next digit (6) is 5 or more
1.239 rounded to units is 1 ... the next digit (2) is less than 5
Rounding to Significant Digits

To round "so many" significant digits, just count digits from left to right, and then round off from there.

Note: if there are leading zeros (such as 0.006), don't count them because they are only there to show how small the number is.
Examples Because ...
1.239 rounded to 3 significant digits is 1.24 ... the next digit (9) is 5 or more
134.9 rounded to 1 significant digit is 100 ... the next digit (3) is less than 5
0.0165 rounded to 2 significant digits is 0.017 ... the next digit (5) is 5 or more

Significant Digit Calculator

(Try increasing or decreasing the number of significant digits. Also try numbers with lots of zeros in front of them like 0.00314, 0.0000314 etc)

Decimals

Decimals
A Decimal Number (based on the number 10) contains a Decimal Point.
Place Value

To understand decimal numbers you must first know about Place Value.

When we write numbers, the position (or "place") of each number is important.

In the number 327:

* the "7" is in the Units position, meaning just 7 (or 7 "1"s),
* the "2" is in the Tens position meaning 2 tens (or twenty),
* and the "3" is in the Hundreds position, meaning 3 hundreds.

Place Value
"Three Hundred Twenty Seven"
keft As we move left, each position is 10 times bigger!
From Units, to Tens, to Hundreds

... and ...
As we move right, each position is 10 times smaller. right
From Hundreds, to Tens, to Units

decimals-tenths

But what if we continue past Units?

What is 10 times smaller than Units?

1/10 ths (Tenths) are!



But we must first write a decimal point,
so we know exactly where the Units position is: tenths
"three hundred twenty seven and four tenths"

And that is a Decimal Number!
Decimal Point

The decimal point is the most important part of a Decimal Number. It is exactly to the right of the Units position. Without it, we would be lost ... and not know what each position meant.

Now we can continue with smaller and smaller values, from tenths, to hundredths, and so on, like in this example:


Large and Small

So, our Decimal System lets us write numbers as large or as small as we want, using the decimal point. Numbers can be placed to the left or right of a decimal point, to indicate values greater than one or less than one.
17.591
The number to the left of the decimal point is a whole number (17 for example)

As we move further left, every number place gets 10 times bigger.

The first digit on the right means tenths (1/10).

As we move further right, every number place gets 10 times smaller (one tenth as big).

Definition of Decimal

The word "Decimal" really means "based on 10" (From Latin decima: a tenth part).

We sometimes say "decimal" when we mean anything to do with our numbering system, but a "Decimal Number" usually means there is a Decimal Point.
Ways to think about Decimal Numbers ...
... as a Whole Number Plus Tenths, Hundredths, etc

You could think of a decimal number as a whole number plus tenths, hundredths, etc:
Example 1: What is 2.3 ?

* On the left side is "2", that is the whole number part.
* The 3 is in the "tenths" position, meaning "3 tenths", or 3/10
* So, 2.3 is "2 and 3 tenths"

Example 2: What is 13.76 ?

* On the left side is "13", that is the whole number part.
* There are two digits on the right side, the 7 is in the "tenths" position, and the 6 is the "hundredths" position
* So, 13.76 is "13 and 7 tenths and 6 hundredths"

... as a Decimal Fraction

Or, you could think of a decimal number as a Decimal Fraction.

A Decimal Fraction is a fraction where the denominator (the bottom number) is a number such as 10, 100, 1000, etc (in other words a power of ten)

So "2.3" would look like this:
23
10

And "13.76" would look like this:
1376
100


... as a Whole Number and Decimal Fraction

Or, you could think of a decimal number as a Whole Number plus a Decimal Fraction.
So "2.3" would look like this:
2 and
3
10

And "13.76" would look like this:
13 and
76
100