Solved Exercises on Prime and Composite Numbers 1 to 100

Master prime and composite numbers from 1 to 100: identifying primes, finding composites, and understanding patterns through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying Prime Numbers 1-20
Exercise 1
List all prime numbers from 1 to 20 and explain why each is prime.
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 20
  2. List all factors of each number
  3. If exactly 2 factors, it's prime
  4. If more than 2 factors, it's composite
Range
1-20
Primes
2,3,5,7,11,13,17,19
Count
8 primes
Step 1: Test each number from 2 to 20

2: Factors are 1, 2 (exactly 2 factors) → PRIME

3: Factors are 1, 3 (exactly 2 factors) → PRIME

4: Factors are 1, 2, 4 (more than 2 factors) → COMPOSITE

5: Factors are 1, 5 (exactly 2 factors) → PRIME

6: Factors are 1, 2, 3, 6 (more than 2 factors) → COMPOSITE

7: Factors are 1, 7 (exactly 2 factors) → PRIME

8: Factors are 1, 2, 4, 8 (more than 2 factors) → COMPOSITE

9: Factors are 1, 3, 9 (more than 2 factors) → COMPOSITE

10: Factors are 1, 2, 5, 10 (more than 2 factors) → COMPOSITE

11: Factors are 1, 11 (exactly 2 factors) → PRIME

12: Factors are 1, 2, 3, 4, 6, 12 (more than 2 factors) → COMPOSITE

13: Factors are 1, 13 (exactly 2 factors) → PRIME

14: Factors are 1, 2, 7, 14 (more than 2 factors) → COMPOSITE

15: Factors are 1, 3, 5, 15 (more than 2 factors) → COMPOSITE

16: Factors are 1, 2, 4, 8, 16 (more than 2 factors) → COMPOSITE

17: Factors are 1, 17 (exactly 2 factors) → PRIME

18: Factors are 1, 2, 3, 6, 9, 18 (more than 2 factors) → COMPOSITE

19: Factors are 1, 19 (exactly 2 factors) → PRIME

20: Factors are 1, 2, 4, 5, 10, 20 (more than 2 factors) → COMPOSITE

Step 2: Compile the list of primes

Prime numbers from 1 to 20: 2, 3, 5, 7, 11, 13, 17, 19

Primes 1-20: 2, 3, 5, 7, 11, 13, 17, 19
Final answer:

The prime numbers from 1 to 20 are: 2, 3, 5, 7, 11, 13, 17, 19. Each has exactly two factors: 1 and itself.

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 Identifying Composite Numbers 21-40
Exercise 2
List all composite numbers from 21 to 40 and provide their factors.
Definition:

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

Range
21-40
Composites
21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40
Primes
23,29,31,37
Step 1: Identify all numbers from 21 to 40

21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40

Step 2: Determine which are composite

21: Factors 1, 3, 7, 21 → COMPOSITE

22: Factors 1, 2, 11, 22 → COMPOSITE

23: Factors 1, 23 → PRIME

24: Factors 1, 2, 3, 4, 6, 8, 12, 24 → COMPOSITE

25: Factors 1, 5, 25 → COMPOSITE

26: Factors 1, 2, 13, 26 → COMPOSITE

27: Factors 1, 3, 9, 27 → COMPOSITE

28: Factors 1, 2, 4, 7, 14, 28 → COMPOSITE

29: Factors 1, 29 → PRIME

30: Factors 1, 2, 3, 5, 6, 10, 15, 30 → COMPOSITE

31: Factors 1, 31 → PRIME

32: Factors 1, 2, 4, 8, 16, 32 → COMPOSITE

33: Factors 1, 3, 11, 33 → COMPOSITE

34: Factors 1, 2, 17, 34 → COMPOSITE

35: Factors 1, 5, 7, 35 → COMPOSITE

36: Factors 1, 2, 3, 4, 6, 9, 12, 18, 36 → COMPOSITE

37: Factors 1, 37 → PRIME

38: Factors 1, 2, 19, 38 → COMPOSITE

39: Factors 1, 3, 13, 39 → COMPOSITE

40: Factors 1, 2, 4, 5, 8, 10, 20, 40 → COMPOSITE

Step 3: List all composite numbers

Composite numbers from 21 to 40: 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40

Composites 21-40: 21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40
Final answer:

The composite numbers from 21 to 40 are: 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40. These have more than two factors each.

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 in Range 41-60
Exercise 3
Determine which numbers from 41 to 60 are prime and which are composite. Show your work.
Definition:

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

Range
41-60
Primes
41,43,47,53,59
Composites
42,44,45,46,48,49,50,51,52,54,55,56,57,58,60
Step 1: Test each number from 41 to 60

41: Only divisible by 1 and 41 → PRIME

42: Divisible by 1, 2, 3, 6, 7, 14, 21, 42 → COMPOSITE

43: Only divisible by 1 and 43 → PRIME

44: Divisible by 1, 2, 4, 11, 22, 44 → COMPOSITE

45: Divisible by 1, 3, 5, 9, 15, 45 → COMPOSITE

46: Divisible by 1, 2, 23, 46 → COMPOSITE

47: Only divisible by 1 and 47 → PRIME

48: Divisible by 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 → COMPOSITE

49: Divisible by 1, 7, 49 → COMPOSITE (7²)

50: Divisible by 1, 2, 5, 10, 25, 50 → COMPOSITE

51: Divisible by 1, 3, 17, 51 → COMPOSITE

52: Divisible by 1, 2, 4, 13, 26, 52 → COMPOSITE

53: Only divisible by 1 and 53 → PRIME

54: Divisible by 1, 2, 3, 6, 9, 18, 27, 54 → COMPOSITE

55: Divisible by 1, 5, 11, 55 → COMPOSITE

56: Divisible by 1, 2, 4, 7, 8, 14, 28, 56 → COMPOSITE

57: Divisible by 1, 3, 19, 57 → COMPOSITE

58: Divisible by 1, 2, 29, 58 → COMPOSITE

59: Only divisible by 1 and 59 → PRIME

60: Divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 → COMPOSITE

Step 2: Compile results

Prime numbers from 41 to 60: 41, 43, 47, 53, 59

Composite numbers from 41 to 60: 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60

Primes 41-60: 41,43,47,53,59, Composites: 42,44,45,46,48,49,50,51,52,54,55,56,57,58,60
Final answer:

Prime numbers from 41 to 60: 41, 43, 47, 53, 59. Composite numbers: 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60.

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

Prime and Composite Numbers 1-100 Summary
\(\text{Prime}: \text{Exactly 2 factors (1 and itself)}\)
Prime Number Definition
Primes 1-100
\(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\)
25 total primes
Composites 1-100
\(4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100\)
74 total composites
Special Cases
\(1\)
Neither prime nor composite
Key definitions:

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

Factor: A number that divides another number evenly (with no remainder)

Divisor: A number that can divide another number without remainder

Prime Factorization: Expressing a number as a product of prime numbers

Factor Pair: Two numbers that multiply to give the original number

Prime/composite identification methods:
  1. List all factors: Find all numbers that divide the target number evenly
  2. Count factors: If exactly 2, it's prime; if more than 2, it's composite
  3. Use divisibility rules: Check for quick division by small numbers
  4. Square root shortcut: Only check divisors up to √n
  5. Check small primes: Test divisibility by 2, 3, 5, 7, etc.
  6. Verify result: Confirm your classification makes sense
Tip 1: All even numbers greater than 2 are composite (divisible by 2).
Tip 2: Numbers ending in 5 (except 5 itself) are composite (divisible by 5).
Tip 3: If a number is divisible by 3, add digits and check if sum is divisible by 3.
Tip 4: Use the square root rule to save time when testing large numbers.
Tip 5: Memorize the first 25 primes (they appear frequently).
Tip 6: Perfect squares are always composite (except 1).
Common errors: Forgetting that 1 is neither prime nor composite, thinking all odd numbers are prime, not checking enough divisors, miscounting factors, confusing factors with multiples.
Key notes: 2 is the only even prime number, prime numbers are building blocks for all other numbers, primes become less frequent as numbers get larger.
Essential facts for 1-100:

• Total primes: 25

• Total composites: 74

• Neither prime nor composite: 1

• 2 is the only even prime number

• Largest prime in range: 97

• Smallest composite: 4

Solution: Exercises 4 to 5
4 Prime Factorization in Range 61-80
Exercise 4
Find the prime factorization of all composite numbers from 61 to 80.
Definition:

Prime factorization: Expressing a number as a product of prime numbers

Range
61-80
Primes
61,67,71,73,79
Composites
62-80 (except primes)
Step 1: Identify composite numbers from 61 to 80

Composites: 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80

Step 2: Find prime factorization for each composite

62 = 2 × 31

63 = 3² × 7

64 = 2⁶

65 = 5 × 13

66 = 2 × 3 × 11

68 = 2² × 17

69 = 3 × 23

70 = 2 × 5 × 7

72 = 2³ × 3²

74 = 2 × 37

75 = 3 × 5²

76 = 2² × 19

77 = 7 × 11

78 = 2 × 3 × 13

80 = 2⁴ × 5

Step 3: Identify prime numbers in range

Prime numbers from 61 to 80: 61, 67, 71, 73, 79

Prime factorizations shown above
Final answer:

Prime factorizations: 62=2×31, 63=3²×7, 64=2⁶, 65=5×13, 66=2×3×11, 68=2²×17, 69=3×23, 70=2×5×7, 72=2³×3², 74=2×37, 75=3×5², 76=2²×19, 77=7×11, 78=2×3×13, 80=2⁴×5

Applied rules:

• Prime Factorization: Repeatedly divide by smallest prime factors

• Factor Count: For prime factorization a^m × b^n, factors = (m+1)(n+1)

• Unique Factorization: Every composite number has a unique prime factorization

5 Patterns in Prime Numbers 81-100
Exercise 5
Identify all prime numbers from 81 to 100 and describe any patterns you notice.
Definition:

Pattern recognition: Observing trends in prime number distribution

Range
81-100
Primes
83,89,97
Pattern
Less frequent
Step 1: Test each number from 81 to 100 for primality

81: Divisible by 3 (8+1=9) → COMPOSITE

82: Even → COMPOSITE

83: Only divisible by 1 and 83 → PRIME

84: Even → COMPOSITE

85: Ends in 5 → COMPOSITE

86: Even → COMPOSITE

87: Divisible by 3 (8+7=15) → COMPOSITE

88: Even → COMPOSITE

89: Only divisible by 1 and 89 → PRIME

90: Even and ends in 0 → COMPOSITE

91: Divisible by 7 (7×13=91) → COMPOSITE

92: Even → COMPOSITE

93: Divisible by 3 (9+3=12) → COMPOSITE

94: Even → COMPOSITE

95: Ends in 5 → COMPOSITE

96: Even → COMPOSITE

97: Only divisible by 1 and 97 → PRIME

98: Even → COMPOSITE

99: Divisible by 3 (9+9=18) → COMPOSITE

100: Even and ends in 0 → COMPOSITE

Step 2: List the prime numbers from 81 to 100

Prime numbers from 81 to 100: 83, 89, 97

Step 3: Describe observed patterns

• Only 3 primes in the last 20 numbers (81-100)

• Primes are becoming less frequent as numbers get larger

• All primes in this range are odd numbers

• The gaps between consecutive primes are getting larger

Primes 81-100: 83,89,97 (3 total)
Final answer:

The prime numbers from 81 to 100 are: 83, 89, 97. The pattern shows that primes become less frequent as numbers increase, with only 3 primes in the last 20 numbers of our range.

Applied rules:

• Prime Density: Primes become less common as numbers get larger

• Even/Odd Rule: All primes > 2 are odd

• Pattern Recognition: Observe trends in number distribution

Complete Guide to Prime and Composite Numbers 1-100
\(\text{Prime}: \text{Exactly 2 factors}\)
Prime Definition
Key definitions:

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

Factor: A number that divides another number evenly with no remainder

Prime Factorization: Expressing a number as a product of prime numbers

Factor Tree: A visual representation of prime factorization

Divisibility Rules: Quick methods to check if one number divides another

Factor Pair: Two numbers that multiply to give the original number

Prime/composite identification methodology:
  1. Initial assessment: Check if number is 1 (neither prime nor composite)
  2. Quick elimination: Even numbers > 2 and multiples of 5 are composite
  3. List all factors: Find all numbers that divide the target evenly
  4. Count factors: Exactly 2 = prime, more than 2 = composite
  5. Use shortcuts: Apply divisibility rules and square root method
  6. Verify: Double-check your classification
Tip 1: The first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Tip 2: To find factors efficiently, test divisors up to the square root of the number.
Tip 3: If a number ends in 0, 2, 4, 6, or 8, it's divisible by 2 (composite if > 2).
Tip 4: For divisibility by 3, add all digits and check if sum is divisible by 3.
Tip 5: Numbers ending in 5 (except 5 itself) are divisible by 5 and thus composite.
Tip 6: Perfect squares (4, 9, 16, 25, 36, 49, 64, 81, 100) are always composite.
Tip 7: Always verify your answer by checking that all factors divide the number evenly.
Tip 8: Primes become less frequent as numbers get larger.
Common errors: Forgetting that 1 is neither prime nor composite, thinking all odd numbers are prime, not checking enough divisors, miscounting factors, confusing factors with multiples, assuming larger numbers are always composite.
Exam preparation: Memorize the first 25 prime numbers, practice divisibility rules, master the square root shortcut, understand that 2 is the only even prime, practice prime factorization, work on comparison problems, recognize patterns in prime distribution.
Essential facts for 1-100:

• Total primes: 25 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97)

• Total composites: 74

• Neither prime nor composite: 1

• 2 is the only even prime number

• Largest prime in range: 97

• Smallest composite: 4

• Primes become less frequent as numbers increase

• Every composite number has at least one prime factor ≤ √n

Prime and Composite Numbers 1-100 Visualization
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100

Analysis: The grid shows all numbers from 1 to 100 with prime numbers highlighted in yellow, composite numbers in blue, and the special number 1 in red.

Statistics: 25 primes, 74 composites, 1 special case (neither prime nor composite)

Questions & Answers

Question: Why is 1 neither prime nor composite?

Answer: The number 1 is special because it has only one factor: itself (1). Let's look at the definitions:

  • Prime number: Must have exactly 2 factors (1 and itself)
  • Composite number: Must have more than 2 factors

Since 1 only has 1 factor, it doesn't meet the criteria for either category. This is why 1 gets its own special classification as "neither prime nor composite."

This definition helps keep mathematical theorems and properties consistent, particularly in prime factorization where every number has a unique prime factorization.

Question: Is 2 a prime number? I thought all even numbers were composite.

Answer: Yes, 2 is definitely a prime number! You're right that 2 is even, but it's a special case:

  • 2 is divisible by 1: 2 ÷ 1 = 2
  • 2 is divisible by 2: 2 ÷ 2 = 1
  • 2 has exactly 2 factors: 1 and 2

So 2 meets the definition of a prime number perfectly. It's actually the smallest prime number and the only even prime number.

The general rule is: "All even numbers greater than 2 are composite," which means 2 is the exception to that rule.

This is important because 2 is needed for prime factorization of even numbers!

Question: How can I quickly tell if a number is prime without listing all factors?

Answer: Here are efficient methods to test primality:

  1. Check divisibility rules: If the number is even, ends in 5, or digit sum is divisible by 3, it's composite
  2. Square root rule: Only test divisors up to √n (where n is your number)
  3. Test small primes: Check divisibility by 2, 3, 5, 7, 11, 13, etc.
  4. Stop early: If you find any divisor other than 1 and the number itself, it's composite

Example for 47: √47 ≈ 6.9, so test 2, 3, 5:

  • 47 is not even → not divisible by 2
  • 4+7=11, not divisible by 3 → not divisible by 3
  • Doesn't end in 5 → not divisible by 5

Since no divisors were found, 47 is prime!

Question: What's the difference between factors and multiples?

Answer: Here's the key difference:

Factors: Numbers that divide evenly into another number

  • Example: Factors of 12 are 1, 2, 3, 4, 6, 12
  • Factors are always smaller than or equal to the number
  • There's a finite number of factors for any number

Multiples: Results of multiplying a number by whole numbers

  • Example: Multiples of 3 are 3, 6, 9, 12, 15, 18...
  • They get larger as you continue the sequence
  • There are infinitely many multiples for any number

Think of it this way: factors go INTO a number, multiples come FROM a number!

When determining if a number is prime or composite, we focus on its factors.

Question: What's the best way to practice identifying prime and composite numbers?

Answer: Here's an effective practice strategy:

  1. Memorize small primes: Know the first 25 prime numbers by heart
  2. Practice divisibility rules: Master quick checks for 2, 3, 5, 9, 10
  3. Use the square root method: Practice testing divisors only up to √n
  4. Work systematically: Always check small primes first (2, 3, 5, 7...)
  5. Create lists: Make charts of primes and composites in different ranges
  6. Take timed challenges: Practice identifying numbers quickly

Daily practice with 10-15 numbers will significantly improve your speed and accuracy!

Remember: Understanding the concepts deeply is more important than just memorizing numbers!