Prime Analysis Slides
Grade 8 • Number Theory & Real Numbers
Module 1.2: Multiplicative Structure
Mathematical Foundations • Divisibility Analysis
Prime vs. Composite Structures
Investigate whole numbers through factor sets, area models, and divisibility algorithms. Discover why prime numbers serve as the fundamental building blocks of arithmetic and modern cryptography.
Concept 1
Factor Set Analysis
Concept 2
Area & Array Modeling
Concept 3
Proof & Justification
Academic Word Bank & Core Definitions
Reference Lexicon
Factor (Divisor)
\(a \mid n\)
An integer \(a\) is a factor of \(n\) if there exists an integer \(b\) such that \(a \cdot b = n\), dividing \(n\) with remainder zero.
Factors of \(20 = \{1, 2, 4, 5, 10, 20\}\)
Area Model
Geometric Representation
A geometric representation of whole numbers as rectangular areas where length and width correspond to factor pairs.
Area of 18: \(1 \times 18\), \(2 \times 9\), or \(3 \times 6\)
Prime Number
Cardinality = 2
An integer \(p > 1\) having exactly two distinct positive divisors: 1 and \(p\).
Primes: \(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31\dots\)
Composite Number
Cardinality ≥ 3
An integer \(c > 1\) that has at least one divisor other than 1 and itself (3 or more factors).
Composites: \(4, 6, 8, 9, 10, 12, 14, 15, 16\dots\)
Fundamental Theorem of Arithmetic: Every composite number has a unique prime factorization.
Geometric Analysis: The Rectangular Model
Dimensional Factorization
Prime Integer: \(n = 13\) PRIME
An integer \(p\) is prime if and only if it admits only the trivial factorization \(1 \times p\).
Only 1 Valid Rectangular Configuration: \(1 \times 13\)
No non-trivial integer dimensions \(a \times b\) exist where \(a, b > 1\).
Factor set: \(F_{13} = \{1, 13\}\) • \(|F_{13}| = 2\)
Composite Integer: \(n = 16\) COMPOSITE
An integer \(c\) is composite if it can be represented by two or more distinct rectangular dimensions.
\(1 \times 16\)
Linear line
\(2 \times 8\)
Rectangle
\(4 \times 4\)
Square array
Factor set: \(F_{16} = \{1, 2, 4, 8, 16\}\) • \(|F_{16}| = 5\)
Geometric takeaway: Prime numbers cannot be configured into non-trivial rectangular blocks.
Guided Testing: The Square Root Divisibility Bound
Testing Protocol: \(p \le \sqrt{n}\)
Model 1: Test \(n = 59\) PRIME
1. Calculate bound: \(\sqrt{59} \approx 7.68\) (test primes \(\le 7\))
2. Test 2, 3, 5: 59 is odd, sum=14, ends in 9 ✗
3. Test 7: \(59 \div 7 = 8\text{ R }3\) ✗
Divisors: only 1 and 59
59 is proven PRIME
Model 2: Test \(n = 91\) COMPOSITE
1. Calculate bound: \(\sqrt{91} \approx 9.53\) (test primes \(\le 7\))
2. Test 2, 3, 5: Not divisible by 2, 3, or 5 ✗
3. Test 7: \(91 \div 7 = 13\) (Remainder 0) ✓
Divisors: 1, 7, 13, 91 (4 factors)
91 is COMPOSITE
Efficiency Principle: You only need to test prime divisors up to \(\sqrt{n}\).
Mathematical Edge Cases & Common Fallacies
Critical Pitfalls
1
The Number 1
Neither Prime Nor Composite
By formal mathematical definition, primes must have exactly two unique positive divisors. 1 has only one divisor (\(|F_1| = 1\)).
Mathematical Classification: Unit
2
The Number 2
The Only Even Prime
2 satisfies the definition (\(F_2 = \{1, 2\}\)). Any even integer \(2k > 2\) contains at least three divisors: 1, 2, and \(2k\).
Even Primes Set: \{ 2 \}
51
Odd ≠ Prime
The Pseudo-Prime Illusion
Numbers such as 51 (\(3 \times 17\)), 87 (\(3 \times 29\)), and 91 (\(7 \times 13\)) appear prime at a glance, but decompose into smaller factors.
Divisibility test: \(\sum \text{digits} \pmod 3\)
Never infer primality purely from odd parity — always execute divisibility verification!
Check for Understanding: Instant Classification
Partner Analysis
CASE 1
43
Factor Set:
\{ 1, 43 \}
PRIME
CASE 2
51
Factor Set:
1, 3, 17, 51
COMPOSITE
CASE 3
79
Factor Set:
\{ 1, 79 \}
PRIME
CASE 4
87
Factor Set:
1, 3, 29, 87
COMPOSITE
Transition to individual practice: Complete the Prime Analysis Worksheet.
Independent Lab
Prime Analysis Worksheet
Prime & Composite Analysis
Grade 8 Mathematics • Number Systems & Divisibility
Name: ___________________
Period: _____
Date: _________
Mathematical Reference & Word Bank
Factor (Divisor): \(a\) divides \(n\) evenly with remainder 0 (\(a \mid n\)).
\(\sqrt{n}\) Bound: Test only prime divisors \(p \le \sqrt{n}\).
Prime Number: \(p > 1\) with exactly 2 factors: 1 and \(p\).
Composite: \(c > 1\) with ≥ 3 distinct factors.
Guided Examples: Testing Protocol (\(p \le \sqrt{n}\))
Worked Solutions
Model A: Test \(n = 29\) PRIME
• Bound: \(\sqrt{29} \approx 5.38\) → Test primes \(\{2, 3, 5\}\). None divide 29.
• Factor set: \(F_{29} = \{1, 29\}\) • Cardinality: \(|F_{29}| = 2\) → PRIME
Model B: Test \(n = 51\) COMPOSITE
• Bound: \(\sqrt{51} \approx 7.14\) → Test \(\{2, 3, 5, 7\}\). Sum \(5+1=6\) → \(3 \times 17 = 51\).
• Factor set: \(F_{51} = \{1, 3, 17, 51\}\) • \(|F_{51}| = 4\) → COMPOSITE
1 Factor Set Determination & Classification Matrix
List all positive factors in ascending order.
| Integer | Complete Factor Set \(\{d_1, d_2, \dots\}\) | Count \(|F_n|\) | Classification (Circle) |
| --- | --- | --- | --- |
| 19 | | | Prime Composite |
| 39 | | | Prime Composite |
| 47 | | | Prime Composite |
| 63 | | | Prime Composite |
| 71 | | | Prime Composite |
| 91 | | | Prime Composite |
2 Check for Understanding
Concept & Example Verification
Part A: True or False
1. The number 1 is classified as a prime number.
True False
2. All odd whole numbers greater than 1 are prime.
True False
Part B: Provide Examples
3. Write an example of a prime number between 40 and 50:
Prime Number:
4. Write an example of an odd composite number between 20 and 40:
Odd Composite:
Grade 8 Mathematics • Number Systems • Prime & Composite Analysis Single-Page Practice Sheet
Prime Analysis Answer Key
Prime & Composite Analysis • Answer Key
Grade 8
Complete Worked Solutions • Factor Breakdowns • Scoring Rubric
Solutions Sheet
✓ Part 1: Factor Set Determination — Solutions Matrix
1 pt factor set, 1 pt count, 1 pt classification (18 pts total)
| Integer | Complete Factor Set \(F_n\) | Count \(|F_n|\) | Classification |
| --- | --- | --- | --- |
| 19 | \{ 1, 19 \} | 2 | PRIME |
| 39 | \{ 1, 3, 13, 39 \} | 4 | COMPOSITE |
| 47 | \{ 1, 47 \} | 2 | PRIME |
| 63 | \{ 1, 3, 7, 9, 21, 63 \} | 6 | COMPOSITE |
| 71 | \{ 1, 71 \} | 2 | PRIME |
| 91 | \{ 1, 7, 13, 91 \} | 4 | COMPOSITE |
✓ Section 2: Check for Understanding — Solutions & Teacher Notes
1 pt per question (4 pts total)
Part A: True or False Keys
1. Number 1 is prime: FALSE
Reason: A prime number requires exactly two distinct positive factors. 1 has only one factor (\(1 \times 1 = 1\)), so it is classified as a unit (neither prime nor composite).
2. All odd numbers > 1 are prime: FALSE
Reason: Many odd numbers are composite because they have additional factors, such as 9 (\(3 \times 3\)), 15 (\(3 \times 5\)), 21 (\(3 \times 7\)), 27 (\(3 \times 9\)), and 91 (\(7 \times 13\)).
Part B: Example Generation Keys
3. Prime between 40 and 50:
Accept any ONE: 41, 43, or 47
Note: 45 is composite (\(5 \times 9\)) and 49 is composite (\(7 \times 7\)). Only 41, 43, and 47 have exactly two factors.
4. Odd composite between 20 and 40:
Accept any ONE: 21, 25, 27, 33, 35, or 39
Sample factors: \(21 = 3 \times 7\); \(25 = 5 \times 5\); \(27 = 3 \times 9\); \(33 = 3 \times 11\); \(35 = 5 \times 7\); \(39 = 3 \times 13\).
Quick Diagnostic: If a student misses Q2 or Q4, they are likely conflating "odd" with "prime." Prompt them to recite 3-times or 5-times tables to build counterexamples.
Grade 8 Mathematics • Solutions Guide • Single-Page Edition Total Assessment Value: 22 Points
