Math 365 Lecture Notes © S. Nite 8/18/2012
Section 4-2
Page 1 of 3
4.2 Prime and Composite Numbers
One method to determine factors of a natural number is to use squares of paper or
cubes to represent the number as a rectangle.
Example: Find the positive factors of 12.
A positive integer with exactly two distinct, positive divisors is a prime number.
A positive integer greater than 1 that has a positive factor other than 1 and itself is a
composite number.
Prime Factorization
In the grade 7 Focal Points, “Students continue to develop their understanding of
multiplication and division and the structure of numbers by determining if a
counting number greater than 1 is a prime, and if it is not, by factoring it into a
product of primes” (p. 19).
An expression of a number as a product of factors is a factorization. A factorization
containing only prime numbers is a prime factorization.
Theorem 4-12 Fundamental Theorem of Arithmetic
Each composite number can be written as a product of primes in one, and only one,
way except for the order of the prime factors in the product.
Note: The primes in the prime factorization of a number are typically listed in
increasing order from left to right. Exponential notation may be used.
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 4-2
Page 2 of 3
Number of Divisors
Example: Find the number of divisors for 24.
The prime factorization can be used with the fundamental counting principle to find
the number of positive divisors.
Theorem 4-13
If p and q are different primes, then pnqm has (n + 1)(m + 1) positive divisors. In
general, if p1, p2, . . . , pk are primes and n1, n2, . . . , nk are whole numbers, then
p1n ⋅ p1n ⋅ ... ⋅ pkn as (n1 + 1)(n2 + 1)…(nk + 1) positive divisors.
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Example: Find the number of divisors for 16,200.
Determine Whether a Number Is Prime
Theorem 4-14
If d is a divisor of n, then
n
is also a divisor of n.
d
Theorem 4-15
If n is composite, then n has a prime factor p such that p2 ≤ n.
Theorem 4-16
If n is an integer greater than 1 and not divisible by any prime p, such that p2 ≤ n,
then n is prime.
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 4-2
Page 3 of 3
One way to find all the primes less than a given number is to use the Sieve of
Eratosthenes.
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More About Primes
There is a whole branch of research and study of primes.
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