A) 15
B) 23
C) 27
D) 33
Prime Number: A number greater than 1 that has exactly two factors: 1 and itself
Composite Number: A number greater than 1 that has more than two factors
- Test each option by finding its factors
- Count the number of factors
- Prime = exactly 2 factors
- Eliminate incorrect options
Factors of 15: 1, 3, 5, 15
Count: 4 factors
Since 4 > 2 → 15 is composite
Eliminate A
Factors of 23: 1, 23
Count: 2 factors
Since exactly 2 factors → 23 is prime
Keep B as a possibility
Factors of 27: 1, 3, 9, 27
Count: 4 factors
Since 4 > 2 → 27 is composite
Eliminate C
Factors of 33: 1, 3, 11, 33
Count: 4 factors
Since 4 > 2 → 33 is composite
Eliminate D
Option B (23) is the correct answer.
23 is a prime number because it has exactly two factors: 1 and 23.
All other options are composite numbers.
• Factor Counting: List all factors systematically
• Elimination Strategy: Process of elimination
• Prime Definition: Exactly 2 factors = Prime
A) 37
B) 41
C) 43
D) 49
Square Root Method: To test if n is prime, only check divisibility by primes up to √n
√37 ≈ 6.1, check primes ≤ 6: 2, 3, 5
37 ÷ 2 = 18.5 (not whole)
37 ÷ 3 = 12.33 (not whole)
37 ÷ 5 = 7.4 (not whole)
No divisors → 37 is prime
√41 ≈ 6.4, check primes ≤ 6: 2, 3, 5
41 ÷ 2 = 20.5 (not whole)
41 ÷ 3 = 13.67 (not whole)
41 ÷ 5 = 8.2 (not whole)
No divisors → 41 is prime
√43 ≈ 6.6, check primes ≤ 6: 2, 3, 5
43 ÷ 2 = 21.5 (not whole)
43 ÷ 3 = 14.33 (not whole)
43 ÷ 5 = 8.6 (not whole)
No divisors → 43 is prime
√49 = 7, check primes ≤ 7: 2, 3, 5, 7
49 ÷ 2 = 24.5 (not whole)
49 ÷ 3 = 16.33 (not whole)
49 ÷ 5 = 9.8 (not whole)
49 ÷ 7 = 7 (whole number!)
So 49 = 7 × 7 → 49 is composite
Option D (49) is the correct answer.
49 is composite because it equals 7 × 7 and has factors {1, 7, 49}.
Options A, B, and C are all prime numbers.
• Square Root Method: Only test up to √n
• Prime Testing: Check divisibility by primes
• Composite Identification: Find any divisor
A) 2
B) 3
C) 4
D) 5
Range Testing: Systematically checking each number in a range for primality
21, 22, 23, 24, 25, 26, 27, 28, 29
21 = 3 × 7 → composite
22 → even → composite
23 → test √23 ≈ 4.8: 23÷2, 23÷3 → no divisors → prime
24 → even → composite
25 = 5² → composite
26 → even → composite
27 = 3³ → composite
28 → even → composite
29 → test √29 ≈ 5.4: 29÷2, 29÷3, 29÷5 → no divisors → prime
Primes between 20 and 30: 23, 29
Count: 2 prime numbers
Option A (2) is the correct answer.
There are exactly 2 prime numbers between 20 and 30: 23 and 29.
All other numbers in this range are composite.
• Systematic Testing: Check each number in sequence
• Quick Elimination: Even numbers >2 are composite
• Counting Strategy: Keep track of primes found
Prime Number: A number greater than 1 that has exactly two factors: 1 and itself
Composite Number: A number greater than 1 that has more than two factors
Factor: A number that divides another number evenly with no remainder
Multiple: The product of a number and any integer
- Read Carefully: Identify what is being asked
- Show Work: Write out your steps for partial credit
- Be Concise: Provide clear, direct answers
- Check Units: Ensure answer format matches question
• Smallest prime number: 2
• Only even prime: 2
• 1 is neither prime nor composite
• Prime numbers are the "building blocks" of all numbers
• Every composite number can be made by multiplying prime numbers
Sample Short Answer Question:
"Is 37 a prime number? Explain your reasoning."
Expected Answer:
Yes, 37 is a prime number. I tested divisibility by primes up to √37 ≈ 6.1. I checked 2, 3, and 5. 37 is not divisible by any of these, so it has exactly two factors: 1 and 37. Therefore, 37 is prime.
Group Formation: Using factors to determine possible arrangements
Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}
These represent all possible group sizes
From the factors {1, 2, 3, 4, 6, 8, 12, 24}:
1: Not prime (only 1 factor)
2: Prime (factors: {1, 2})
3: Prime (factors: {1, 3})
4: Composite (factors: {1, 2, 4})
6: Composite (factors: {1, 2, 3, 6})
8: Composite (factors: {1, 2, 4, 8})
12: Composite (factors: {1, 2, 3, 4, 6, 12})
24: Composite (factors: {1, 2, 3, 4, 6, 8, 12, 24})
Prime group sizes: 2 and 3
If groups of 2: 24 ÷ 2 = 12 groups
If groups of 3: 24 ÷ 3 = 8 groups
12 groups × 2 students = 24 students ✓
8 groups × 3 students = 24 students ✓
• 12 groups of 2 students
• 8 groups of 3 students
The teacher has two possible arrangements:
1) 12 groups with 2 students each
2) 8 groups with 3 students each
Both 2 and 3 are prime numbers, satisfying the condition that each group must have a prime number of students.
• Factor Analysis: Use factors to determine arrangements
• Prime Identification: Apply prime number definitions
• Verification: Check that arrangements work
Mathematical Reasoning: Using logical arguments to explain mathematical concepts
Even numbers are integers divisible by 2
Any even number > 2 can be written as 2k where k ≥ 2
For any even number n = 2k where k ≥ 2:
n has at least these factors: 1, 2, k, n
If k > 2, then 2k has at least 4 factors: {1, 2, k, 2k}
This means n has more than 2 factors
For n = 2: k = 1
2k = 2 × 1 = 2
Factors of 2: {1, 2}
Exactly 2 factors → 2 is prime
Example 1: 4 = 2 × 2, factors: {1, 2, 4} → composite
Example 2: 6 = 2 × 3, factors: {1, 2, 3, 6} → composite
Example 3: 8 = 2 × 4, factors: {1, 2, 4, 8} → composite
Example 4: 10 = 2 × 5, factors: {1, 2, 5, 10} → composite
For any even number n > 2, n = 2k where k ≥ 2
This means n has factors 1, 2, k, and n
Since n > 2, we have k ≥ 2, so k ≠ 1 and k ≠ n
Therefore n has at least 4 factors, making it composite
Only exception: n = 2, which has exactly 2 factors
2 is the only even prime number because:
1) Every even number > 2 is divisible by 2
2) If an even number n > 2, then n = 2k where k ≥ 2
3) This means n has at least 4 factors: {1, 2, k, n}
4) Since n has more than 2 factors, n is composite
5) Only 2 itself has exactly 2 factors: {1, 2}
Therefore, 2 is the only even number that is prime.
• General Proof: Use algebraic reasoning
• Counterexamples: Show why other numbers don't work
• Logical Structure: Build argument step by step
Prime Number: A natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself
Composite Number: A natural number greater than 1 that has more than two distinct positive divisors
Prime Factorization: Expressing a composite number as a unique product of prime numbers raised to powers
Factor: A number that divides another number evenly with no remainder
Multiple: The product of a number and any integer
- Direct Factorization: List all factors and count
- Square Root Method: Efficient prime testing
- Divisibility Rules: Quick elimination of composites
- Prime Factorization: Advanced number analysis
- Application Problems: Real-world context
• All primes > 3 are of the form 6n±1
• The sum of digits test: if sum divisible by 3, number divisible by 3
• If last digit is 0 or 5, number divisible by 5
• Prime factorization is unique for every number
• There are 25 primes less than 100
• Square root method: only test up to √n
Question Types:
- Multiple Choice: Quick identification and testing
- Short Answer: Brief explanations and calculations
- Extended Response: Detailed reasoning and problem-solving
- Application: Real-world scenarios
- Proof/Reasoning: Mathematical justification
Each category tests different levels of understanding!