The Greatest Common Factor and Least Common Multiple come in handy when adding, subtracting, multiplying, dividing or reducing fractions. The GCF is the biggest factor that two numbers have in common. The LCM is the smallest multiple that two numbers have in common. This song introduces students to the GCF and LCM and demonstrates how to find them.

When we’re messing with fractions and need to reduce,

The GCF is the tool that we use.

The Greatest Common Factor that both numbers share,

Who invented it? I have no idea.

But to reduce

^{32}⁄

_{40},

We find all the factors of each, then look for the

Biggest one in common, that’s the GCF,

Divide them both by it and see what’s left.

The factors for 32:

1 and 32, 2 and 16, 4 and 8, yep that’s who.

Factors for 40:

1 and 40, 2 and 20, 4 and 10, 5 and 8, that’s them.

Factors in common: 1, 2, 4, 8,

So 8 is the GCF, but wait!

Divide the top and bottom by the GCF,

You get

^{4}⁄

_{5}, that’s reduced, you’re set!

GCF, that’s my Greatest Common Factor,

The biggest number going into both, move it backwards.

LCM, that’s my Least Common Multiple,

The lowest number that both numbers can go into. (x2)

Factors are smaller, multiples are bigger,

You get them when you multiply, get the picture?

6, 12 and 18 are multiples of 6,

And there are way more, in fact it’s infinite.

The smallest multiple that numbers share,

Is the Least Common Multiple, let’s be clear.

To find the LCM of 6 and 8,

Find the first few multiples of six, ok?

6, 12, 18, 24,

We could keep going, there are plenty more.

Now let’s do the same for 8,

And wait until we see a match on the plate:

8, 16, 24 hold up, that’s him!

24 must be the LCM.

You could also do this by factoring primes,

But that’s another lesson kid, for another time.

GCF, that’s my Greatest Common Factor,

The biggest number going into both, move it backwards.

LCM, that’s my Least Common Multiple,

The lowest number that both numbers can go into. (x2)

What is the GCF? It stands for greatest common factor, and it's useful when simplifying fractions. Common factors are whole numbers that divide evenly into both of the numbers in question, and the greatest common factor is the largest of those whole numbers.

^{
32
}
⁄
_{
40
}
is the sort of answer you might get after adding or multiplying fractions, but you may need to simplify your answer as a final step. Use the GCF to reduce
^{
32
}
⁄
_{
40
}
to its simplest form.

Simple, right? Now let's try
^{
36
}
⁄
_{
81
}
.

Factors for 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Factors for 81: 1, 3, 9, 27, 81

Factors in common: 1, 3, 9

So 9 is the greatest common factor. Divide the top and bottom by 9 to get
^{
4
}
⁄
_{
9
}
, and you have the simplified fraction.

Think of multiples as the opposite of factors: they are the bigger numbers that a smaller number can go into evenly. We get multiplies when we multiply a number by another other whole number. Don't forget that any number is always a multiple of itself, because it equals itself when multiplied by one.

So multiples of 4 are 4, 8, 12, 16, 20, 24, 28, etc.

The multiplies of 100 would be 100, 200, 300, 400, etc.

Multiples of any number are infinite because you can always multiply that number by one more number. And then one more... For example, there is no limit to multiples of six.

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204...

It just keeps going and going!

Two numbers will share a lot of multiples, but the Least Common Multiple, or smallest multiple that two numbers have in common, is what we're after. The LCM comes in handy when adding and subtracting fractions, since you have to make the denominator the same for those operations.

So if you're trying to subtract
^{
3
}
⁄
_{
8
}
from
^{
5
}
⁄
_{
6
}
, you use the LCM 24 as the denominator for both fractions.

^{
20
}
⁄
_{
24
}
–
^{
12
}
⁄
_{
24
}
=
^{
8
}
⁄
_{
24
}

You can then use the GCF (in this case, 8) to reduce the fraction:
^{
8
}
⁄
_{
24
}
=
^{
1
}
⁄
_{
3
}

How about
^{
3
}
⁄
_{
5
}
+
^{
1
}
⁄
_{
4
}
? What is the LCM of 5 and 4?

Factoring primes (also called prime factorization) is another way to find the LCM and the GCF. You do this by listing all the prime numbers that have to be multiplied together to equal the number you're looking at.