Solved Exercises on Prime and Composite Numbers Assessment

Master prime and composite numbers: identifying primes, finding composites, and understanding factors through these 5 detailed exercises.

Assessment Solution: Exercises 1 to 3
1 Prime Number Assessment
Exercise 1
Determine if 19 is a prime number and explain your reasoning.
Definition:

Prime number: A number greater than 1 that has exactly two factors: 1 and itself

Prime identification method:
  1. Test each number from 2 to √19 ≈ 4.4
  2. List all factors of the number
  3. Count the total number of factors
  4. If there are exactly 2 factors, it's prime
Number
19
Factors
1, 19
Type
Prime
Step 1: Find all factors of 19

We need to find all numbers that divide 19 evenly (with no remainder)

19 ÷ 1 = 19 (remainder 0) ✓

19 ÷ 2 = 9 remainder 1 (not a factor)

19 ÷ 3 = 6 remainder 1 (not a factor)

19 ÷ 4 = 4 remainder 3 (not a factor)

Step 2: Count the factors

Only 1 and 19 divide 19 evenly

Total factors: 2

Step 3: Determine if prime or composite

Since 19 has exactly 2 factors (1 and 19), it is a prime number

19 is PRIME
Final answer:

Yes, 19 is a prime number because it has exactly two factors: 1 and 19

Applied rules:

• Prime Definition: Exactly two factors: 1 and the number itself

• Factor Testing: Check divisibility by all numbers less than the square root

• Counting Factors: Prime numbers have exactly 2 factors

2 Composite Number Assessment
Exercise 2
Determine if 28 is a composite number and list all its factors.
Definition:

Composite number: A number greater than 1 that has more than two factors

Number
28
Factors
1,2,4,7,14,28
Type
Composite
Step 1: Find all factors of 28

We need to find all numbers that divide 28 evenly (with no remainder)

28 ÷ 1 = 28 (remainder 0) ✓

28 ÷ 2 = 14 (remainder 0) ✓

28 ÷ 4 = 7 (remainder 0) ✓

28 ÷ 7 = 4 (remainder 0) ✓

28 ÷ 14 = 2 (remainder 0) ✓

28 ÷ 28 = 1 (remainder 0) ✓

Step 2: List all factors in order

Factors of 28: 1, 2, 4, 7, 14, 28

Step 3: Count the factors

Total factors: 6 (more than 2)

Step 4: Determine if prime or composite

Since 28 has more than 2 factors, it is a composite number

28 is COMPOSITE with factors: 1, 2, 4, 7, 14, 28
Final answer:

Yes, 28 is a composite number because it has more than two factors: 1, 2, 4, 7, 14, 28

Applied rules:

• Composite Definition: More than two factors

• Factor Pairs: When you find one factor, you also find its pair

• Systematic Approach: Check divisors up to the square root of the number

3 Finding Prime or Composite Assessment
Exercise 3
Determine if 31 is prime or composite. Show your work.
Definition:

Prime number: Has exactly two factors (1 and itself)

Number
31
Factors
1, 31
Type
Prime
Step 1: Test divisibility by small prime numbers

We only need to check divisors up to √31 ≈ 5.6, so check 2, 3, 5

31 ÷ 2 = 15 remainder 1 (not divisible)

31 ÷ 3 = 10 remainder 1 (not divisible)

31 ÷ 5 = 6 remainder 1 (not divisible)

Step 2: Identify the factors

Only 1 and 31 divide 31 evenly

Factors: 1, 31

Step 3: Determine if prime or composite

Since 31 has exactly 2 factors, it is prime

31 is PRIME
Final answer:

31 is a prime number because it has exactly two factors: 1 and 31

Applied rules:

• Square Root Rule: Only check divisors up to √n

• Efficient Testing: Check small primes first (2, 3, 5, 7...)

• Prime Definition: Exactly two factors confirms primality

Assessment Rubric & Scoring Guide
\(a_n = a_1 + (n-1)d\)
Arithmetic Sequence Formula
Prime
\(2, 3, 5, 7, 11, 13...\)
d is common difference
Geometric
\(a_n = a_1 \times r^{n-1}\)
r is common ratio
Square Numbers
\(a_n = n^2\)
n is the position
Key definitions:

Arithmetic Sequence: Each term increases by a constant difference (d)

Geometric Sequence: Each term is multiplied by a constant ratio (r)

Pattern: A regular, repeated design or sequence

Term: Each individual number in a sequence

Position: The place of a term in the sequence (1st, 2nd, 3rd...)

Assessment scoring criteria:
  1. Correct identification: Accurately determine if prime or composite (4 points)
  2. Complete work shown: Demonstrate all steps clearly (3 points)
  3. Proper reasoning: Explain the logic behind the answer (2 points)
  4. Accurate factorization: List all factors correctly (2 points)
  5. Verification: Check that the answer is correct (1 point)
Tip 1: Always start by finding differences between consecutive terms.
Tip 2: If differences aren't constant, try dividing consecutive terms.
Tip 3: Look for familiar number patterns like squares, cubes, or primes.
Tip 4: If differences of differences are constant, it's a quadratic pattern.
Common errors: Forgetting to check if the pattern holds for all terms, misidentifying arithmetic vs geometric sequences.
Key notes: Practice recognizing common sequences like Fibonacci, triangular numbers, and prime numbers.
Essential formulas:

• Arithmetic: \(a_n = a_1 + (n-1)d\)

• Geometric: \(a_n = a_1 \times r^{n-1}\)

• Square numbers: \(a_n = n^2\)

• Cube numbers: \(a_n = n^3\)

• Triangular numbers: \(a_n = \frac{n(n+1)}{2}\)

Exercise 1: 9/10 points
Exercise 2: 10/10 points
Exercise 3: 9/10 points
Exercise 4: 8/10 points
Exercise 5: 10/10 points
Total Score: 46/50 points (92%)
Grading Rubric:
A (90-100%): Excellent understanding with minimal errors
B (80-89%): Good understanding with minor errors
C (70-79%): Satisfactory understanding with some errors
D (60-69%): Needs improvement with significant errors
F (Below 60%): Requires major review and re-teaching
Assessment Solution: Exercises 4 to 5
4 Complex Pattern Assessment
Exercise 4
Find the rule and the 10th term of the sequence: 2, 5, 10, 17, 26, ...
Definition:

Quadratic pattern: A sequence where the differences of differences are constant

Sequence
2, 5, 10, 17, 26, ...
First Differences
3, 5, 7, 9, ...
Rule
n² + 1
Step 1: Find the first differences

5 - 2 = 3, 10 - 5 = 5, 17 - 10 = 7, 26 - 17 = 9

First differences: 3, 5, 7, 9 (not constant)

Step 2: Find the second differences

5 - 3 = 2, 7 - 5 = 2, 9 - 7 = 2

Second differences: 2, 2, 2 (constant)

Step 3: Identify the pattern

Since second differences are constant, it's a quadratic pattern

Terms: 2=1²+1, 5=2²+1, 10=3²+1, 17=4²+1, 26=5²+1

Rule: n² + 1

Step 4: Find the 10th term

Apply rule: 10² + 1 = 100 + 1 = 101

Rule: n² + 1, 10th term: 101
Final answer:

The pattern rule is n² + 1, and the 10th term is 101

Applied rules:

• Quadratic Patterns: When second differences are constant

• Difference Analysis: Find differences until a constant appears

• Pattern Recognition: Look for squares, cubes, or other familiar forms

5 Fibonacci Pattern Assessment
Exercise 5
Find the 9th term of the sequence: 1, 1, 2, 3, 5, 8, 13, 21, ...
Definition:

Fibonacci sequence: Each term is the sum of the two preceding terms

Sequence
1, 1, 2, 3, 5, 8, 13, 21, ...
Rule
aₙ = aₙ₋₁ + aₙ₋₂
9th Term
34
Step 1: Analyze the relationship between terms

1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, 8+13=21

Step 2: Identify the recursive rule

Each term equals the sum of the two preceding terms

Step 3: Continue the sequence

8th term: 13 + 21 = 34

Wait, that's not right. We already have the 8th term as 21.

9th term: 13 + 21 = 34

Step 4: Verify the pattern

Check that each term is indeed the sum of the two before it

9th term: 34
Final answer:

The 9th term of the sequence is 34

Applied rules:

• Fibonacci Pattern: Each term = sum of two preceding terms

• Recursive Relations: Terms depend on previous terms

• Sequential Calculation: Build the sequence term by term

Complete Assessment Guide to Number Patterns
\(a_n = a_1 + (n-1)d\)
Arithmetic Pattern Formula
Key definitions:

Pattern: A regular, repeated design or sequence of numbers, shapes, or events

Arithmetic Sequence: Each term increases by a constant difference (d)

Geometric Sequence: Each term is multiplied by a constant ratio (r)

Algebraic Expression: A mathematical phrase with variables, numbers, and operations

Pattern Rule: A formula that describes how to generate any term in a sequence

Recursive Rule: Each term depends on one or more previous terms

Explicit Rule: Direct formula to calculate any term using its position

Assessment methodology:
  1. Read the problem: Understand what is being asked
  2. Identify the sequence type: Arithmetic, geometric, or special pattern
  3. Find the pattern rule: Determine the mathematical relationship
  4. Select the appropriate formula: Use the correct sequence formula
  5. Substitute values: Replace variables with known numbers
  6. Calculate: Perform the mathematical operations
  7. Verify: Check that your answer is reasonable
Tip 1: Always start by finding the differences between consecutive terms.
Tip 2: If differences aren't constant, try dividing consecutive terms to check for geometric sequences.
Tip 3: For complex patterns, look for second differences or higher-order patterns.
Tip 4: Look for familiar number sets: squares (1,4,9,16...), cubes (1,8,27...), or triangular numbers (1,3,6,10...).
Tip 5: If the pattern seems recursive, consider if each term depends on previous terms.
Tip 6: Always verify your answer by checking if it fits the established pattern.
Common errors: Misidentifying arithmetic vs geometric sequences, forgetting to check if the rule holds for all terms, making calculation errors when substituting values, confusing explicit and recursive rules, not verifying the final answer.
Exam preparation: Memorize formulas for arithmetic and geometric sequences, practice identifying various pattern types, develop systematic approaches to pattern recognition, understand when to use explicit vs recursive rules, practice substitution in algebraic expressions.
Essential formulas:

• Arithmetic: \(a_n = a_1 + (n-1)d\)

• Geometric: \(a_n = a_1 \times r^{n-1}\)

• Square numbers: \(a_n = n^2\)

• Cube numbers: \(a_n = n^3\)

• Triangular numbers: \(a_n = \frac{n(n+1)}{2}\)

• Fibonacci: \(a_n = a_{n-1} + a_{n-2}\)

Patterns and Algebra Visualization
Exercise 6: Comparing Different Patterns
Compare different pattern types:
\(f_1(n) = 2n + 1\) (arithmetic)
\(f_2(n) = 2^n\) (geometric)
\(f_3(n) = n^2\) (quadratic)

Analysis: The chart shows how different pattern rules grow over time.

  • \(f_1(n) = 2n + 1\): Linear growth (arithmetic sequence)
  • \(f_2(n) = 2^n\): Exponential growth (geometric sequence)
  • \(f_3(n) = n^2\): Quadratic growth (square numbers)

Questions & Answers

Question: I'm having trouble telling the difference between arithmetic and geometric sequences. Can you explain how to identify each type?

Answer: Great question! Here's how to distinguish between them:

  • Arithmetic Sequence: Find the difference between consecutive terms. If it's the same number each time, it's arithmetic.
  • Geometric Sequence: Divide consecutive terms. If you get the same number each time, it's geometric.

Examples:

  • Arithmetic: 3, 7, 11, 15... (difference is +4 each time)
  • Geometric: 2, 6, 18, 54... (ratio is ×3 each time)

Remember: Arithmetic involves addition/subtraction, geometric involves multiplication/division.

Question: When I see a sequence like 1, 4, 9, 16..., how do I know it's square numbers and not another pattern?

Answer: Recognizing square numbers follows a specific process:

  • Check if each number is a perfect square: √1=1, √4=2, √9=3, √16=4
  • Notice that the square roots form a simple counting sequence: 1, 2, 3, 4...
  • The pattern is n² where n is the position in the sequence

Additional verification: The differences between terms are consecutive odd numbers (3, 5, 7...), which is unique to square numbers.

Other helpful patterns: Cube numbers (1, 8, 27, 64...), triangular numbers (1, 3, 6, 10...), and Fibonacci sequence (1, 1, 2, 3, 5...).

Question: Sometimes I get confused when a pattern has alternating numbers. How do I handle those?

Answer: For alternating patterns, use the separation method:

  • Separate the sequence into two subsequences: one for odd positions, one for even positions
  • Find the pattern for each subsequence independently
  • Combine them back together

Example: In 2, 5, 4, 7, 6, 9...

  • Odd positions (1st, 3rd, 5th...): 2, 4, 6... (even numbers)
  • Even positions (2nd, 4th, 6th...): 5, 7, 9... (odd numbers)

This approach breaks complex patterns into simpler, more manageable parts!

Question: How can I check if my answer is correct when working with number patterns?

Answer: Here are effective methods:

  • Apply the rule: Use your identified pattern to generate the given terms and check if they match
  • Continue the sequence: Generate a few more terms beyond what's asked to ensure consistency
  • Check relationships: Verify that differences, ratios, or other relationships hold throughout

For example, if you found the pattern is "add 3", apply it to the first term: 5 + 3 = 8, which matches the second term ✓

Always double-check your work by confirming the pattern works for ALL given terms, not just some.

Question: What's the best way to practice number patterns to get better at recognizing them quickly?

Answer: Here's a systematic approach to improve pattern recognition:

  1. Start with basics: Master arithmetic (+2, +3, +5...) and geometric (×2, ×3, ×10...) sequences
  2. Memorize common patterns: Squares (1,4,9,16...), cubes (1,8,27,64...), triangular numbers (1,3,6,10...)
  3. Practice systematically: Always check differences first, then ratios
  4. Work backwards: Given the rule, generate the sequence to understand the reverse process
  5. Use visual aids: Draw dots, blocks, or charts to visualize patterns

Daily practice with 5-10 different patterns will significantly improve your recognition speed and accuracy!

Remember: Pattern recognition improves with exposure and practice!