Math Study Guide
Primes
A prime number is a number divisible by only 1 and itself (the number 1 itself is not
considered prime). For example, 17 is prime because it is divisible by only 1 and 17.
The first few primes, in increasing order, are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
43, 47, 53, .
To determine whether a number is prime, it is not necessary to check whether it is
divisible by every number less than itself. It is only necessary to check whether it is
divisible by any prime up to half its value. One might do the following computations to
check whether 91 is prime:
1.
2.
3.
4.
Is 91 divisible by 2? No, it does not end with an even number.
Is 91 divisible by 3? No, 9 + 1 = 10 , which is not divisible by 3.
Is 91 divisible by 5? No, 91 does not end with 0 or 5.
Is 91 divisible by 7? Yes! 91/7 = 13 .
Therefore, 91 is not prime
Composites
A composite number is a number that has at least one factor besides 1 and itself, or any
number which is not prime (although the number 1 is neither prime nor composite). For
example, 91 is a composite number. To determine whether a number is composite,
determine whether or not it is prime. If it is prime, it is not composite; if it is not prime,
then it is composite
.
Factors
"Factors" are the numbers you multiply together to get another number:
A factor of a number divides the number. For example, 2, 3, and 4 are all factors of 12.
To determine whether a number is a factor of another number, we generally use the
divisibility rules.
.
.
Prime Factorization
"Prime Factorization" is finding which prime numbers multiply together to make the
original number:
What are the prime factors of 12?
It is best to start working from the smallest prime number, which is 2, so let's check:
12 ÷ 2 = 6
Yes, it divided evenly by 2. We have taken the first step!
But 6 is not a prime number, so we need to go further. Let's try 2 again:
6÷2=3
Yes, that worked also. And 3 is a prime number, so we have the answer:
12 = 2 × 2 × 3
As you can see, every factor is a prime number, so the answer must be right.
Note: 12 = 2 × 2 × 3 can also be written using exponents as 12 = 22 × 3
Common Factor
After you have worked out the factors of two or more numbers:
Example:
The factors of 12 are 1, 2, 3, 4, 6 and 12
The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
Then the common factors are those that are found in both numbers:
•
•
Notice that 1,2,3 and 6 appear in both lists?
So, the common factors of 12 and 30 are: 1, 2, 3 and 6
It is a common factor when it is a factor of two or more numbers.
(It is then "common to" those numbers.)
Greatest Common Factor
It is simply the largest of the common factors. In our previous example, the largest of
the common factors is 15, so the Greatest Common Factor of 15, 30 and 105 is 15
The "Greatest Common Factor" is the largest of the common factors (of two or more
numbers)
Finding the Greatest Common Factor
•
•
•
find all factors of both numbers
then select the ones that are common to both, and
then choose the greatest.
Least Common Multiple.
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.