5.2: Number Theory

Last updated
Save as PDF

Page ID: 51008

Amy Lagusker
College of the Canyons

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Definition: Factors and Multiples

Let \(mn = p\), then \(m\) and \(n\) are factors of \(p\) and \(p\) is a multiple of \(m\) and \(n\)

Hints about Factors and Multiples

Factors are always smaller than the given number, whereas multiples are always bigger than the given number.

Example \(\PageIndex{1}\)

Find Factors and Multiples of 12

Solution

Factors of 12: 1, 2, 3, 4, 6, & 12
Multiples of 12: 12, 24, 36, 48, 60, …

Partner Activity 1

List all the factors and the first four multiples of 30.

Partner Activity 2 - Finding Primes

Below are the numbers from 0 to 99.
Cross out 0 and 1 (neither prime nor composite) and circle 2 (the first prime)
Cross out all multiples of 2.
Circle 3 (prime) and cross out all multiples of 3.
Circle 5 (prime) and cross out all multiples of 5.
Continue this exercise until each number is either crossed out or circled.
Write all your circled primes below.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

Primes and Composites

Definition: Prime number

Any natural number, which has no factors other than 1 and itself.

Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, …

Definition: Composite Number

Any natural number, which has a factor other than 1 and itself.

Examples: 4, 6, 8, 9, 10, 12, 14, 15, …

Definition: Relatively Prime

Two or more numbers with no factors in common.

Examples: 7 and 8 or 15 and 4

Partner Activity 3

Categorize the following as Prime, Composite or Neither: 0, 1, 2, and any negative number

Prime Factorization (Factor Tree)

Partner Activity 4

Write the prime factorization:

Practice Problems

List all the factors.

List the first four multiplies.

List the prime numbers.

Between 20 and 40
Between 60 and 80
Between 120 and 150

List the composite numbers.

Between 20 and 40
Between 60 and 80
Between 120 and 150

Write the prime factorization.