1. Section 5.1
Number Theory

2. What You Will Learn
Upon completion of this section, you will be able to:
Classify natural numbers as prime or composite.
Understand the rules of divisibility for the numbers 2, 3, 4, 5, 6, 8, 9, and 10.
Write the prime factorization of a natural number.
Determine the greatest common divisor of a group of natural numbers.
Determine the least common multiple of a group of natural numbers.
Understand the nature of prime numbers.

3. Number Theory The study of numbers and their properties.
The numbers we use to count are called counting numbers, or natural numbers, denoted by N.
N = {1, 2, 3, 4, 5, …}

4. Factors
The natural numbers that are multiplied together are called factors of the product.
A natural number may have many factors The factors of 18 are 1, 2, 3, 6, 9 and 18.

5. Divisors
If a and b are natural numbers, we say that a is a divisor of b or a divides b, symbolized a | b if the quotient of b divided by a has a remainder of 0.
If a divides b, then b is divisible by a.

6. Prime and Composite Numbers
A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1.
A composite number is a natural number that is divisible by a number other than itself and 1.
The number 1 is neither prime nor composite; it is called a unit.

7. Rules of Divisibility Divisible by 2 3 4 5 Test The number is even.
The sum of the digits of the number is divisible by 3.
The number formed by the last two digits of the number is divisible by 4.
The number ends in 0 or 5.
Example
924; even
924; = 15 and 15 is divisible by 3
924; 24
is divisible by 4
265; ends in 5

8. Rules of Divisibility 290; ends in 0 The number ends in 0. 10
837; = 18 and 18 is divisible by 9
The sum of the digits of the number is divisible by 9. 9
5824;
824 is divisible by 8
The number formed by the last three digits of the number is divisible by 8. 8
924; divisible by both 2 and 3
The number is divisible by both 2 and 3. 6
Example
Test
Divisible by

9. The Fundamental Theorem of Arithmetic
Every composite number can be expressed as a unique product of prime numbers.
This unique product is referred to as the prime factorization of the number.

10. Finding Prime Factorizations Method 1: Branching
Select any two numbers whose product is the number to be factored.
If the factors are not prime numbers, continue factoring each number until all numbers are prime.

11. Example 2: Prime Factorization by Branching
Write 1500 as a product of primes.
Solution or
Thus, the prime factorization of
1500 = 2 • 2 • 3 • 5 • 5 • 5 = 22 • 3 • 53

12. Method 2: Division
1. Divide the given number by the smallest prime number by which it is divisible.
2. Place the quotient under the given number.
3. Divide the quotient by the smallest prime number by which it is divisible and again record the quotient.
4. Repeat this process until the quotient is a prime number.

13. Example 2: Prime Factorization by Division
Write 1500 as a product of prime numbers.
Solution 2
1500 2
750 3
375 5
125 5 25 5
1500 = 2 • 2 • 3 • 5 • 5 • 5 = 22 • 3 • 53

14. Greatest Common Divisor
The greatest common divisor (GCD) of a set of natural numbers is the largest natural number that divides (without remainder) every number in that set.

15. To Determine the GCD of Two or More Numbers
1. Determine the prime factorization of each number.
2. List each prime factor with smallest exponent that appears in each of the prime factorizations.
3. Determine the product of the factors found in Step 2.

16. Example 4: Using Prime Factorization to Determine the GCD
Determine the GCD of 54 and 90.
Solution
54 = 2 • 33
90 = 2 • 32 • 5
Prime factors with smallest exponents that appear in each factorization
2 and 32
The GCD is 2 • 32 = 18.

17. Least Common Multiple
The least common multiple (LCM) of a set of natural numbers is the smallest natural number that is divisible (without remainder) by each element of the set.

18. To Determine the LCM of Two or More Numbers
1. Determine the prime factorization of each number.
2. List each prime factor with the greatest exponent that appears in any of the prime factorizations.
3. Determine the product of the factors found in Step 2.

19. Example 6: Using Prime Factorization to Determine the LCM
Determine the LCM of 54 and 90.
Solution
54 = 2 • 33
90 = 2 • 32 • 5
Prime factors with greatest exponents that appear in either factorization
2, 33 and 5
The LCM is 2 • 33 • 5 = 270.

20. More About Prime Numbers
More than 2000 years ago, the Greek mathematician Euclid proved that there is no largest prime number.
Mathematicians, however, continue to strive to find larger and larger prime numbers.

21. Mersenne Primes
Marin Mersenne (1588–1648), a seventeenth-century monk, found that numbers of the form 2n – 1 are often prime numbers when n is a prime number
Numbers of the form 2n – 1 are referred to as Mersenne primes.

22. Mersenne Primes
The first 10 Mersenne primes occur when n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89
The first time the expression does not generate a prime number, for prime number n, is when n is 11

23. Mersenne Primes
Largest prime discovered January 25, 2013 by Curtis Cooper, a professor at the University of Central Missouri.
257,885,161 – 1
It is 17,425,170 digits long
Written in 12-point font it is almost 23 miles long!

24. Fermat Numbers
Pierre de Fermat (1601 – 1665) conjectured that each number of the form , now referred to as a Fermat number, was prime for each natural number n.
A conjecture is a hypothesis that has not been proved or disproved.

25. Fermat Numbers
In 1732, Leonhard Euler proved that for n = 5, was a composite number, thus disproving Fermat’s conjecture.
Since Euler’s time, mathematicians have been able to evaluate only the sixth, seventh, eighth, ninth, tenth, and eleventh Fermat numbers and each of these numbers has been shown to be composite.

26. Goldbach’s Conjecture
In 1742, Christian Goldbach conjectured that every even number greater than or equal to 4 can be represented as the sum of two (not necessarily distinct) prime numbers.
For example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 5 + 5, 12 =
It remains unproven to this day.

27. Twin Prime Conjecture
Twin primes are primes of the form p and p + 2 (3 and 5, 11 and 13).
The conjecture states that there are an infinite number of pairs of twin primes.
The largest known twin primes are of the form 3,756,801,695,685 • ± 1.
They were found by the efforts of two research groups: Twin Prime Search and PrimeGrid on December 25, 2011.