Properties of Integers:
I. Prime Factorization
a. divisors
'd' is a divisor of 'a' if there is another integer c such that
c * d = a
Example: 5 is a divisor of 15 since 3 * 5 = 15
Example: 3 is a divisor of 15 since 5 * 3 =
15
Example: 2 is not a divisor of 15 since there is no
number to
multiply 2 by to equal 15
b. prime number
if p is an number other than 0, 1, or -1, and it has no divisors
other than 1 or -1, then p is called a prime number
Example: 20 is not a prime number since 2 and 5 are
divisors of 20
Example: 3, 5, 7, 11, 13, 17, 19 are prime numbers
FUNDAMENTAL
THEOREM OF ARITHMETIC:
Any positive
integer greater than one can be expressed as a product of positive prime
integers in only one way (not including order).
note: these prime numbers
are called factors
note: the product can
be a single number, the prime number itself.
Example: 12 = 2 * 6, 6 is not prime, so we have 2 * (2 * 3) =
2*2*3
Example: 13 = 13
Example: 42 = 2 * 21 = 2 * (3 * 7) = 2*3*7
Example: 81 = 3 * 27 = 3 * (3 * 9) = 3*3*(3 * 3) = 3*3*3*3
c. Exponent
It will become cumbersome to write repeated factors of larger numbers, so for
notational convenience, if a prime is repeated, we can write it as that prime
with a superscript that is the number of times that prime is repeated.
Example: 3 * 3 * 3 * 3 will be written as 34
Example: 2 * 2 = 22
Example: 13 =
131 (no superscript implies a superscript of 1)
Example: 648 = 2 * 324 = 2 * 2 *
162
= 2 * 2 * 2 * 81 = 2*2*2*3*3*3*3 = 23*34
So, 23*34
MEANS 2*2*2*3*3*3*3
Hints for finding the prime
factorization:
1. Start with the prime
divisors 2, 3, 5, 7, 11, 13, etc, in order, until you reach the square
root of the number (more on this later)
2. Recognize that if 2
divides the last digit in the number, 2 is a factor
3. #2 is also
true for 5
4. If you add up the
digits in a number and 3 divides that sum then 3 divides that number.
Example:
12 (last digit is 2, 2 divides 2 so start with 2)
12 = 2 * 6 (look at 6, 2 divides 6 so we have)
12 = 2 * ( 2 * 3 ) ( look at 3, 3 is prime, we're done)
12 = 22 * 3
Example:
30 ( 2 divides 0 )
30 = 2 * 15 ( look at 15, 1 + 5
= 6 and 3 divides 6)
30 = 2 * ( 3 * 5 ) (look at 5, 5 is prime, we're
done )
30 = 2 * 3 * 5
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