Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lecture, we saw that numbers which divide evenly into a natural number
are said to be that number's "factors".
For instance, since three divides evenly into six, we say that three is a factor of six.
Numbers that have exactly two factors, themselves and one, are called "prime numbers".
Numbers greater than one which are not prime are "composite" numbers,
and all composite numbers can be broken down into a product of prime numbers.
In fact, one of the most important theorems of mathematics,
called the "Fundamental Theorem of Arithmetic",
states that any integer greater than one can be written as a unique product of prime numbers.
So we can think of prime numbers as the elementary building blocks of composite numbers.
The process of breaking down a composite number into all its prime factors is called "factoring".
To factor a number, we must find the primes,
which when multiplied together, produce that number.
Take as an example 420,
which is a product of the prime numbers two, two, three, five, and seven.
We can think of these factors as the elementary building blocks of 420.
To factor a number we need to know if it is a prime or composite.
We can determine this by testing each prime number
to see if it divides evenly into the number.
We start with the smallest prime, two.
In this example, since 420 is an even number,
we know it is divisible by two.
Therefore two is a factor.
Dividing 420 by two leaves 210,
so 210 is also a factor.
But is it prime or composite?
If it is prime then we can stop.
We have found all the prime factors.
However, 210 is even,
so we can factor out another two.
We now have two factors of two, times 105.
If 105 is prime, then we can stop.
We know that two is not a factor of 105
because 105 is not even.
But it may be divisible by the next prime, three.
Remembering the trick from our last lecture,
if the sum of a number’s digits
is divisible by three,
then that number is divisible by three.
Since the digits of 105 add up to six,
and six is divisible by three,
we know that 105 is divisible by three.
Therefore, three is a factor.
We now have the factors two, times two, times three, times 35.
We must now determine if 35 is prime or composite.
Since 35 is not divisible by three,
we test the next prime number, five.
35 is divisible by five, so five is a factor.
Now we have the factors two, times two, times three, times five, times seven.
Since seven is a prime number,
we are finished factoring.
We have decomposed the composite number 420 into its prime factors.
Now let’s try an example with a number which is not as easy to factor, 849.
Once again, we start by testing
the smallest prime number, two.
Since 849 is not even,
we know that two is not a factor,
so we go on to the next prime number, three.
Since the digits of 849 add up to 21 which is divisible by three, three is factor.
849 divided by three is 283,
so 283 is also a factor.
But is it prime or composite?
If it is prime then we can stop.
We have found all the prime factors.
Its digits add up to thirteen,
which is not divisible by three.
So we go on to the next prime number, five.
Since 283 does not end in five or zero,
five is not a factor.
So we test the next prime number, seven.
Here is where a calculator comes in handy.
Factors must divide evenly into the number being factored.
Therefore, dividing by any valid factor must result in an integer.
However, if we divide 283 by seven,
we don’t get an integer,
so seven is ruled out as a factor.
The same is true for the next two primes,
eleven, and thirteen.
Notice that as we divide by larger numbers,
the results keep getting smaller.
When we divide by the next prime, seventeen,
the result is smaller than seventeen,
so we can stop testing. Why?
Since factors must divide evenly,
the result must be an integer,
and that integer will also be a factor.
We know that there are no more factors
smaller than seventeen
because we have already tested them all.
Therefore, we can stop testing.
Since we can find no factors of 283,
it must be a prime number.
So three and 283 are the only two factors of 849.
As we have seen, factoring a number involves breaking it down into prime numbers,
whose product is that number.
Factoring is an extremely important tool in mathematics.
In the next few lectures, we will see that factoring is the key
to many types of arithmetic operations involving fractions.