12.1: Number Theory
- Page ID
- 132932
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Number theory is a vast topic in mathematics. For the purposes of this course, we will concern ourselves with only a few topics. The first such topic is the relationship between multiplication and division among the nonnegative integers.
Let \(m\), \(n\), and \(p\) be nonnegative integers such that \(mn = p\). Then \(m\) and \(n\) are factors of \(p\), and \(p\) is a multiple of \(m\) and \(n\)
Let's use these words with specific values.
- Find all the positive integer factors of 12.
- Find at least 5 positive integer multiples of 12.
Solution
- Factors of 12: 1, 2, 3, 4, 6, & 12
- Multiples of 12: 12, 24, 36, 48, 60, …
Now, try one on your own.
- List all the positive integer factors for 30. Hint: there are eight of them!
- List at least 5 positive integers multiples of 30.
Primes and Composites
Any natural number (positive integer) that has exactly two natural number factors is called a prime number. Those two factors are 1 and itself.
Some examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, … Notice that 0 is not a positive number, and even if we allowed 0 to count, it has infinitely many factors because 0 times any integer equals 0. So, 0 is definitely not prime. Similarly, 1 has only one positive integer factor, which is itself. So 1 is also not prime.
Any natural number (positive integer), which has a positive integer factor other than 1 and itself is called a composite number.
Some examples of composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, … Notice that this time we do not include 0 simply by definition since it is not positive.
Relatively Prime Pairs of Numbers
Two or more positive integers whose only common positive integer factor is 1 are relatively prime.
Show 7 and 8 are relatively prime.
Solution
The positive integer factors of 7 are 1 and 7 (tit is prime).
The positive integer factors of 8 are 1, 2, 4, and 8.
The only common number between them is 1. So, 7 and 8 are relatively prime.
Now try one on your own.
Show 4 and 15 are relatively prime
Can you find other pairs of relatively prime positive integers?