In earlier lectures, we introduced the Natural Numbers N = {1, 2, 3, 4, 5, ...}. In this lecture, we are going to discuss some interesting things that can be done with these numbers.
Divisibility
If a and b are natural numbers, a is divisible by b if the operation results in a remainder of 0.
Tests for Divisibility
A natural number is divisible by
2 if the last digit is an even number (0, 2, 4, 6, 8, etc.)
3 if the sum of its digits is divisible by 3
4 if the last two digits form a number that is divisible by 4
5 if the last digit is 0 or 5
6 if the number is divisible by both 2 and 3
7 if the division has no remainder
8 if the last three digits form a number divisible by 8
9 if the sum of its digits is divisible by 9
10 if the last digit is 0
Prime Numbers
A prime number is a natural number greater than 1 that only has itself and 1 as factors. For example, 2, 3, 5, 7, 11, 13, 17, etc. are prime numbers.
Composite Numbers
A composite number is a natural number greater than 1 that is divisible by a number other than itself and 1.
THE FUNDAMENTAL THEOREM OF ARITHMETIC
Every composite number can be expressed as a product of prime numbers in exactly one unique way.
How to Write Composite Numbers as a Product of Prime Factors
This method will be explained using two examples.
Let's write the numbers 80 and 12 as a product of prime factors. We will utilize the Test of Divisibility to help us.
Basically, we start with the smallest prime number after the number 1 and utilize it until it no longer divides into the remainder. Then we proceed to the next higher prime number and repeat the process until the multiplier equals 1.
Factor 80 into prime factors.
2 is the smallest prime number and it is a factor of 80:
80 = 2 40
2 is the smallest prime number and it is a factor of the multiplier 40:
40 = 2 20
2 is the smallest prime number and it is a factor of the multiplier 20:
20 = 2 10
2 is the smallest prime number and it is a factor of the multiplier 10:
10 = 2 5
2 is no longer a factor of the multiplier 5. 3 is the next higher prime number and it is also not a factor of the multiplier 5. As a matter of fact, 5 is a prime number!
5 = 5 1
We are done with the prime factorization process when the multiplier equals 1!
This shows that 80 can be written as a product of prime factors as follows:
80 = 2 2 2 2 5
We can use exponents to show the repeated prime factors. That is, .
Factor 12 into prime factors.
2 is the smallest prime number and it is a factor of 12:
12 = 2 6
2 is the smallest prime number and it is a factor of the multiplier 6:
6 = 2 3
2 is the smallest prime number and it is not a factor of the multiplier 3. As a matter of fact, 3 is a prime number!
3 = 3 1
We are done with the prime factorization process when the multiplier equals 1!
This shows that 12 can be written as a product of prime factors as follows:
12 = 2 2 3
We can use exponents to show the repeated prime factors. That is, .
The Greatest Common Divisor
The greatest common divisor of two or more natural numbers is the largest number that divides into all of the numbers. Some pairs of numbers have only 1 as their greatest common divisor. These numbers are called relatively prime. For example 24 and 7 are relatively prime whereas 24 and 30 have a greatest common divisor of 6.
Finding the Greatest Common Divisor
This method will be explained using the examples from above. There we found that and .
Actually, wealready did the first step, which is writing the two numbers in terms of their product of prime factors.
In the second step, we will select ONLY the prime factors that are common to BOTH numbers. However, if indicated we will select the one with the smaller exponent. In our case, we will select and NOT . We will NOT select 3 and 5 because they are not factors of both numbers.
We find that the greatest common divisor of the numbers 80 and 12 is or 4.
The Least Common Multiple
The least common multiple of two or more natural numbers is the smallest number that is divisible by all of the numbers.
Finding the Least Common Multiple
This method will be explained using the examples from above. There we found that and .
Again, we already did the first step, which is writing the two numbers in terms of their product of prime factors.
In the second step, we will select EVERY prime factor that occurs on both numbers raised to the greatest power to which it occurs. In our case, we will select and also 3 and 5.
