Prime Number: A number greater than 1 that has exactly two factors: 1 and itself
- Start with 1 and check if it divides the number evenly
- Check 2, 3, 4, 5, up to the number itself
- List all numbers that divide evenly
- Count the total factors
- If exactly 2 factors exist, it's prime
41 ÷ 1 = 41 remainder 0 → 1 is a factor
41 ÷ 2 = 20 remainder 1 → 2 is not a factor
41 ÷ 3 = 13 remainder 2 → 3 is not a factor
41 ÷ 4 = 10 remainder 1 → 4 is not a factor
41 ÷ 5 = 8 remainder 1 → 5 is not a factor
41 ÷ 6 = 6 remainder 5 → 6 is not a factor
41 ÷ 7 = 5 remainder 6 → 7 is not a factor
41 ÷ 8 = 5 remainder 1 → 8 is not a factor
41 ÷ 9 = 4 remainder 5 → 9 is not a factor
41 ÷ 10 = 4 remainder 1 → 10 is not a factor
41 ÷ 11 = 3 remainder 8 → 11 is not a factor
41 ÷ 12 = 3 remainder 5 → 12 is not a factor
41 ÷ 13 = 3 remainder 2 → 13 is not a factor
41 ÷ 14 = 2 remainder 13 → 14 is not a factor
41 ÷ 15 = 2 remainder 11 → 15 is not a factor
41 ÷ 16 = 2 remainder 9 → 16 is not a factor
41 ÷ 17 = 2 remainder 7 → 17 is not a factor
41 ÷ 18 = 2 remainder 5 → 18 is not a factor
41 ÷ 19 = 2 remainder 3 → 19 is not a factor
41 ÷ 20 = 2 remainder 1 → 20 is not a factor
41 ÷ 21 = 1 remainder 20 → 21 is not a factor
41 ÷ 22 = 1 remainder 19 → 22 is not a factor
41 ÷ 23 = 1 remainder 18 → 23 is not a factor
41 ÷ 24 = 1 remainder 17 → 24 is not a factor
41 ÷ 25 = 1 remainder 16 → 25 is not a factor
41 ÷ 26 = 1 remainder 15 → 26 is not a factor
41 ÷ 27 = 1 remainder 14 → 27 is not a factor
41 ÷ 28 = 1 remainder 13 → 28 is not a factor
41 ÷ 29 = 1 remainder 12 → 29 is not a factor
41 ÷ 30 = 1 remainder 11 → 30 is not a factor
41 ÷ 31 = 1 remainder 10 → 31 is not a factor
41 ÷ 32 = 1 remainder 9 → 32 is not a factor
41 ÷ 33 = 1 remainder 8 → 33 is not a factor
41 ÷ 34 = 1 remainder 7 → 34 is not a factor
41 ÷ 35 = 1 remainder 6 → 35 is not a factor
41 ÷ 36 = 1 remainder 5 → 36 is not a factor
41 ÷ 37 = 1 remainder 4 → 37 is not a factor
41 ÷ 38 = 1 remainder 3 → 38 is not a factor
41 ÷ 39 = 1 remainder 2 → 39 is not a factor
41 ÷ 40 = 1 remainder 1 → 40 is not a factor
41 ÷ 41 = 1 remainder 0 → 41 is a factor
Yes, 41 is a prime number because it has exactly two factors: 1 and 41.
• Prime Definition: Must have exactly two factors
• Factor Testing: Check divisibility systematically
• Counting: Only 2 factors means it's prime
Composite Number: A number greater than 1 that has more than two factors
48 ÷ 1 = 48 remainder 0 → 1 is a factor
48 ÷ 2 = 24 remainder 0 → 2 is a factor
48 ÷ 3 = 16 remainder 0 → 3 is a factor
48 ÷ 4 = 12 remainder 0 → 4 is a factor
48 ÷ 5 = 9 remainder 3 → 5 is not a factor
48 ÷ 6 = 8 remainder 0 → 6 is a factor
48 ÷ 7 = 6 remainder 6 → 7 is not a factor
48 ÷ 8 = 6 remainder 0 → 8 is a factor
48 ÷ 9 = 5 remainder 3 → 9 is not a factor
48 ÷ 10 = 4 remainder 8 → 10 is not a factor
48 ÷ 11 = 4 remainder 4 → 11 is not a factor
48 ÷ 12 = 4 remainder 0 → 12 is a factor
48 ÷ 13 = 3 remainder 9 → 13 is not a factor
48 ÷ 14 = 3 remainder 6 → 14 is not a factor
48 ÷ 15 = 3 remainder 3 → 15 is not a factor
48 ÷ 16 = 3 remainder 0 → 16 is a factor
48 ÷ 17 = 2 remainder 14 → 17 is not a factor
48 ÷ 18 = 2 remainder 12 → 18 is not a factor
48 ÷ 19 = 2 remainder 10 → 19 is not a factor
48 ÷ 20 = 2 remainder 8 → 20 is not a factor
48 ÷ 21 = 2 remainder 6 → 21 is not a factor
48 ÷ 22 = 2 remainder 4 → 22 is not a factor
48 ÷ 23 = 2 remainder 2 → 23 is not a factor
48 ÷ 24 = 2 remainder 0 → 24 is a factor
48 ÷ 25 = 1 remainder 23 → 25 is not a factor
48 ÷ 26 = 1 remainder 22 → 26 is not a factor
48 ÷ 27 = 1 remainder 21 → 27 is not a factor
48 ÷ 28 = 1 remainder 20 → 28 is not a factor
48 ÷ 29 = 1 remainder 19 → 29 is not a factor
48 ÷ 30 = 1 remainder 18 → 30 is not a factor
48 ÷ 31 = 1 remainder 17 → 31 is not a factor
48 ÷ 32 = 1 remainder 16 → 32 is not a factor
48 ÷ 33 = 1 remainder 15 → 33 is not a factor
48 ÷ 34 = 1 remainder 14 → 34 is not a factor
48 ÷ 35 = 1 remainder 13 → 35 is not a factor
48 ÷ 36 = 1 remainder 12 → 36 is not a factor
48 ÷ 37 = 1 remainder 11 → 37 is not a factor
48 ÷ 38 = 1 remainder 10 → 38 is not a factor
48 ÷ 39 = 1 remainder 9 → 39 is not a factor
48 ÷ 40 = 1 remainder 8 → 40 is not a factor
48 ÷ 41 = 1 remainder 7 → 41 is not a factor
48 ÷ 42 = 1 remainder 6 → 42 is not a factor
48 ÷ 43 = 1 remainder 5 → 43 is not a factor
48 ÷ 44 = 1 remainder 4 → 44 is not a factor
48 ÷ 45 = 1 remainder 3 → 45 is not a factor
48 ÷ 46 = 1 remainder 2 → 46 is not a factor
48 ÷ 47 = 1 remainder 1 → 47 is not a factor
48 ÷ 48 = 1 remainder 0 → 48 is a factor
Yes, 48 is a composite number because it has more than two factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
• Composite Definition: Must have more than two factors
• Factor Pairs: 1×48=48, 2×24=48, 3×16=48, 4×12=48, 6×8=48
• Counting: More than 2 factors means it's composite
Classification: Determining if each number is prime or composite by finding its factors
Factors: 1, 43 → Prime (exactly 2 factors)
Factors: 1, 3, 5, 9, 15, 45 → Composite (6 factors)
Factors: 1, 47 → Prime (exactly 2 factors)
Factors: 1, 7, 49 → Composite (3 factors)
Factors: 1, 3, 17, 51 → Composite (4 factors)
Prime: 43, 47 | Composite: 45, 49, 51
Prime numbers: 43, 47 | Composite numbers: 45, 49, 51
• Systematic Analysis: Examine each number individually
• Factor Counting: Determine the exact number of factors
• Classification Rules: 2 factors = prime, more than 2 = composite
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
Factors: Numbers that divide evenly into another number without leaving a remainder
- Check if number > 1: Prime/composite only applies to numbers greater than 1
- Find all factors: Systematically check divisibility from 1 up to the number
- Count factors: Tally the total number of factors found
- Classify: 2 factors = prime, more than 2 = composite
• 1 is neither prime nor composite
• 2 is the smallest and only even prime number
• All primes greater than 2 are odd numbers
• The first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
• Every composite number has at least one prime factor
Word problems: Real-world situations requiring knowledge of factors and divisibility
Factors of 53: 1, 53
Option 1: 1 row of 53 marbles (1×53)
Option 2: 53 rows of 1 marble each (53×1)
Since 53 has exactly 2 factors (1 and 53), it's prime
James can arrange the marbles in only 2 ways: 1 row of 53 marbles or 53 rows of 1 marble each
James can arrange his 53 marbles in 1 row of 53 marbles or 53 rows of 1 marble each. The number 53 is prime because it has exactly two factors: 1 and 53.
• Prime Application: Prime numbers have limited arrangement options
• Real-world connection: Arranging objects requires understanding factors
• Verification: Confirm that arrangements use all marbles
Range analysis: Systematically checking numbers within a specified range
51, 52, 53, 54, 55, 56, 57, 58, 59
51 = 3 × 17 → composite
52 = 4 × 13 → composite
Factors: 1, 53 → prime
54 = 6 × 9 → composite
55 = 5 × 11 → composite
56 = 7 × 8 → composite
57 = 3 × 19 → composite
58 = 2 × 29 → composite
Factors: 1, 59 → prime
The prime numbers between 50 and 60 are: 53 and 59
• Systematic approach: Check each number in the range individually
• Even number rule: All even numbers > 2 are composite
• Verification: Double-check each classification
Prime Number Problems: Problems focusing on identifying numbers with exactly two factors
Composite Number Problems: Problems focusing on identifying numbers with more than two factors
Classification Problems: Problems that require distinguishing between prime and composite numbers
- Problem 1 (Prime): Take number 47 → factors are 1 and 47 → exactly 2 factors → prime
- Problem 2 (Composite): Take number 50 → factors are 1, 2, 5, 10, 25, 50 → 6 factors → composite
- Problem 3 (Pattern): Notice that all even numbers > 2 are composite
- Problem 4 (Application): Prime numbers limit how items can be arranged equally
• 2 is the only even prime number
• All primes > 2 are odd numbers
• Numbers ending in 5 (except 5) are composite
• Every number > 1 is either prime or composite
• Prime factorization is unique for each number