Prime number: A number greater than 1 that has exactly two factors: 1 and itself. A prime number cannot be divided evenly by any other numbers except 1 and itself.
- Check if the number is greater than 1
- Test if it can be divided evenly by any number from 2 up to its square root
- If it can be divided evenly by any of these numbers, it's composite
- If it cannot be divided evenly by any of these numbers, it's prime
Yes, 17 > 1, so it could be prime
√17 ≈ 4.1, so we test divisibility by 2, 3, and 4
17 ÷ 2 = 8 R 1 (not divisible)
17 ÷ 3 = 5 R 2 (not divisible)
17 ÷ 4 = 4 R 1 (not divisible)
The only numbers that divide 17 evenly are 1 and 17
Factors: 1, 17
Since 17 has exactly two factors (1 and 17), it is a prime number
Yes, 17 is a prime number because it has exactly two factors: 1 and 17.
• Prime definition: Exactly two factors: 1 and itself
• Testing efficiency: Only check divisors up to √n
• Factor counting: Count all numbers that divide evenly
Composite number: A number greater than 1 that has more than two factors. It can be divided evenly by numbers other than 1 and itself.
Yes, 24 > 1, so it could be composite
Test divisibility by numbers 1 to 24:
24 ÷ 1 = 24 (factors: 1, 24)
24 ÷ 2 = 12 (factors: 2, 12)
24 ÷ 3 = 8 (factors: 3, 8)
24 ÷ 4 = 6 (factors: 4, 6)
24 ÷ 5 = 4 R 4 (not a factor)
24 ÷ 6 = 4 (already found)
Factors: 1, 2, 3, 4, 6, 8, 12, 24
Since 24 has more than two factors (8 factors total), it is a composite number
Yes, 24 is a composite number because it has more than two factors: 1, 2, 3, 4, 6, 8, 12, 24.
• Composite definition: More than two factors
• Factor pairing: If n ÷ a = b, then both a and b are factors
• Systematic testing: Test divisors up to n/2
Prime factorization: Writing a composite number as a product of prime numbers. Every composite number can be uniquely expressed as a product of primes.
Write 36 at the top of the tree
36 = 4 × 9 (both are factors of 36)
4 = 2 × 2 (both 2s are prime)
9 = 3 × 3 (both 3s are prime)
36 = 2 × 2 × 3 × 3 = 2² × 3²
The prime factorization of 36 is 2² × 3² or 2 × 2 × 3 × 3.
• Prime factorization: Express number as product of primes only
• Factor tree method: Break down composite numbers recursively
• Unique factorization: Every number has one unique prime factorization
Number Classification
Definition: Has exactly 2 factors
Examples: 2, 3, 5, 7, 11, 13
Special: 2 is the only even prime
Definition: Has more than 2 factors
Examples: 4, 6, 8, 9, 10, 12
Special: Can be factored into primes
Number: 1 (has only 1 factor)
Number: 0 (has infinite factors)
Note: These don't fit either category
Prime number: A whole number greater than 1 that has exactly two factors: 1 and itself
Composite number: A whole number greater than 1 that has more than two factors
Factor: A number that divides another number evenly
Prime factorization: Expressing a number as a product of prime numbers
- Check number: Ensure it's greater than 1
- Find factors: List all numbers that divide it evenly
- Count factors: Two factors = prime, more than two = composite
- Verify: Double-check by multiplication
• Prime rule: Exactly 2 factors (1 and itself)
• Composite rule: More than 2 factors
• Factor test: If a number divides evenly, both divisor and quotient are factors
• Efficiency rule: Only check divisors up to √n for primality
Prime Number: A whole number greater than 1 that has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc.
Composite Number: A whole number greater than 1 that has more than two factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc.
Factor: A number that divides another number evenly without leaving a remainder. For example, factors of 12 are 1, 2, 3, 4, 6, and 12.
Prime Factorization: The process of expressing a composite number as a product of prime numbers. For example, 12 = 2 × 2 × 3 = 2² × 3.
Prime Definition: A number is prime if it has exactly two distinct positive divisors: 1 and itself.
Composite Definition: A number is composite if it has more than two distinct positive divisors.
Number 1 Exception: The number 1 is neither prime nor composite because it has only one factor.
Fundamental Theorem: Every integer greater than 1 is either prime or can be uniquely expressed as a product of primes.
Identifying Prime Numbers:
- Check if the number is greater than 1
- Test divisibility by prime numbers up to the square root of the number
- If none divide evenly, the number is prime
- If any divide evenly, the number is composite
Identifying Composite Numbers:
- Check if the number is greater than 1
- Find at least one factor besides 1 and itself
- List all factors systematically
- If more than two factors exist, it's composite
Prime Factorization (Factor Tree Method):
- Start with the composite number
- Find any two factors (not necessarily prime)
- Continue factoring each non-prime number
- Stop when all branches end in prime numbers
- Write as a product of primes
Simple Example: Is 5 prime? Factors of 5: 1, 5. Only 2 factors, so 5 is prime.
Intermediate Example: Is 15 composite? Factors of 15: 1, 3, 5, 15. More than 2 factors, so 15 is composite.
Advanced Example: Prime factorization of 60: 60 = 4 × 15 = 2² × 3 × 5.
Tips:
- Memorize the first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23
- Remember that 2 is the only even prime number
- For large numbers, only test divisors up to the square root
- Use divisibility rules to quickly identify factors
- When prime factorizing, start with the smallest prime factors
Common Mistakes:
- Thinking 1 is a prime number (it's neither prime nor composite)
- Believing that all odd numbers are prime
- Forgetting that 2 is prime (the only even prime)
- Mistaking composite numbers for primes
- Not completing factor trees to reach all primes
Memory Aid for Prime: "Prime" sounds like "first" - primes are the "building blocks" of all numbers.
Memory Aid for Composite: "Composite" means "made up of parts" - it's composed of multiple prime factors.
Quick Check: If a number ends in 0, 2, 4, 6, or 8 (except 2), it's composite. If it ends in 5 (except 5), it's composite.
Special Cases: Remember that 0 and 1 are neither prime nor composite.
Questions & Answers
Question: Why isn't 1 considered a prime number? It can only be divided by 1 and itself.
Answer: This is a common question! While 1 only has one factor (itself), mathematicians decided that 1 should be in its own category. Here's why:
The fundamental theorem of arithmetic states that every number greater than 1 can be expressed as a unique product of primes. If 1 were prime, this uniqueness would break down because we could write numbers like 6 = 2 × 3, or 6 = 1 × 2 × 3, or 6 = 1 × 1 × 2 × 3, etc.
So 1 is classified as "neither prime nor composite" to maintain the important properties of prime numbers in mathematics. This makes the system cleaner and more consistent.
Question: My child is struggling to find all the factors of a number systematically. How can I help them organize their work?
Answer: Use a systematic approach with factor pairs! Here's how:
- Start with 1: Write "1 × [number] = [number]"
- Check 2: See if 2 divides the number evenly, write the pair
- Continue sequentially: Check 3, 4, 5, etc.
- Stop at √n: Once you pass the square root, you'll just repeat pairs
- Organize: List factors in order from smallest to largest
For example, for 24: 1×24, 2×12, 3×8, 4×6. Then stop because 5 doesn't divide 24 evenly. Factors: 1, 2, 3, 4, 6, 8, 12, 24.
Question: How can I help my students understand prime factorization better?
Answer: Use visual models and connect to prior knowledge:
- Factor trees: Show how numbers "break down" into smaller parts
- Concrete materials: Use tiles or blocks to show how composite numbers can be arranged in multiple rectangular arrays
- Prime as "atoms:" Compare primes to atoms that build all other numbers
- Multiple paths: Show that different factor trees lead to the same prime factorization
Emphasize that prime factorization is like finding the "DNA" of a number - it reveals the fundamental building blocks that make up that number. This helps students see why it's always the same regardless of the path taken.