Solved Worksheets on Prime and Composite Numbers in Grade 4

Master prime and composite numbers through worksheets: identifying primes, finding composites, understanding divisibility, and recognizing number properties through these 5 detailed exercises.

Worksheet Solutions: 1 to 3
1 Worksheet: Prime Identification
Worksheet 1
Circle all the prime numbers in this list: 23, 25, 27, 29, 31, 33, 35, 37
Definition:

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

Method:
  1. Take each number in the list one by one
  2. Find all factors of that number
  3. Count the total number of factors
  4. If exactly 2 factors exist, circle it (it's prime)
  5. If more than 2 factors exist, leave it (it's composite)
Numbers
23, 25, 27, 29, 31, 33, 35, 37
Factors
List each
Types
P, C, C, P, P, C, C, P
Step 1: Analyze 23

Factors of 23: 1, 23 → Prime (exactly 2 factors)

Step 2: Analyze 25

Factors of 25: 1, 5, 25 → Composite (3 factors)

Step 3: Analyze 27

Factors of 27: 1, 3, 9, 27 → Composite (4 factors)

Step 4: Analyze 29

Factors of 29: 1, 29 → Prime (exactly 2 factors)

Step 5: Analyze 31

Factors of 31: 1, 31 → Prime (exactly 2 factors)

Step 6: Analyze 33

Factors of 33: 1, 3, 11, 33 → Composite (4 factors)

Step 7: Analyze 35

Factors of 35: 1, 5, 7, 35 → Composite (4 factors)

Step 8: Analyze 37

Factors of 37: 1, 37 → Prime (exactly 2 factors)

Step 9: Circle the primes

Circle: 23, 29, 31, 37

Step 10: Leave the composites

Leave uncircled: 25, 27, 33, 35

Step 11: Verify 23

23 ÷ 1 = 23 remainder 0 → 1 is a factor

23 ÷ 2 = 11 remainder 1 → 2 is not a factor

23 ÷ 3 = 7 remainder 2 → 3 is not a factor

23 ÷ 4 = 5 remainder 3 → 4 is not a factor

23 ÷ 5 = 4 remainder 3 → 5 is not a factor

23 ÷ 6 = 3 remainder 5 → 6 is not a factor

23 ÷ 7 = 3 remainder 2 → 7 is not a factor

23 ÷ 8 = 2 remainder 7 → 8 is not a factor

23 ÷ 9 = 2 remainder 5 → 9 is not a factor

23 ÷ 10 = 2 remainder 3 → 10 is not a factor

23 ÷ 11 = 2 remainder 1 → 11 is not a factor

23 ÷ 12 = 1 remainder 11 → 12 is not a factor

23 ÷ 13 = 1 remainder 10 → 13 is not a factor

23 ÷ 14 = 1 remainder 9 → 14 is not a factor

23 ÷ 15 = 1 remainder 8 → 15 is not a factor

23 ÷ 16 = 1 remainder 7 → 16 is not a factor

23 ÷ 17 = 1 remainder 6 → 17 is not a factor

23 ÷ 18 = 1 remainder 5 → 18 is not a factor

23 ÷ 19 = 1 remainder 4 → 19 is not a factor

23 ÷ 20 = 1 remainder 3 → 20 is not a factor

23 ÷ 21 = 1 remainder 2 → 21 is not a factor

23 ÷ 22 = 1 remainder 1 → 22 is not a factor

23 ÷ 23 = 1 remainder 0 → 23 is a factor

Step 12: Verify 25

25 ÷ 1 = 25 remainder 0 → 1 is a factor

25 ÷ 2 = 12 remainder 1 → 2 is not a factor

25 ÷ 3 = 8 remainder 1 → 3 is not a factor

25 ÷ 4 = 6 remainder 1 → 4 is not a factor

25 ÷ 5 = 5 remainder 0 → 5 is a factor

25 ÷ 6 = 4 remainder 1 → 6 is not a factor

25 ÷ 7 = 3 remainder 4 → 7 is not a factor

25 ÷ 8 = 3 remainder 1 → 8 is not a factor

25 ÷ 9 = 2 remainder 7 → 9 is not a factor

25 ÷ 10 = 2 remainder 5 → 10 is not a factor

25 ÷ 11 = 2 remainder 3 → 11 is not a factor

25 ÷ 12 = 2 remainder 1 → 12 is not a factor

25 ÷ 13 = 1 remainder 12 → 13 is not a factor

25 ÷ 14 = 1 remainder 11 → 14 is not a factor

25 ÷ 15 = 1 remainder 10 → 15 is not a factor

25 ÷ 16 = 1 remainder 9 → 16 is not a factor

25 ÷ 17 = 1 remainder 8 → 17 is not a factor

25 ÷ 18 = 1 remainder 7 → 18 is not a factor

25 ÷ 19 = 1 remainder 6 → 19 is not a factor

25 ÷ 20 = 1 remainder 5 → 20 is not a factor

25 ÷ 21 = 1 remainder 4 → 21 is not a factor

25 ÷ 22 = 1 remainder 3 → 22 is not a factor

25 ÷ 23 = 1 remainder 2 → 23 is not a factor

25 ÷ 24 = 1 remainder 1 → 24 is not a factor

25 ÷ 25 = 1 remainder 0 → 25 is a factor

Step 13: Verify 27

27 ÷ 1 = 27 remainder 0 → 1 is a factor

27 ÷ 2 = 13 remainder 1 → 2 is not a factor

27 ÷ 3 = 9 remainder 0 → 3 is a factor

27 ÷ 4 = 6 remainder 3 → 4 is not a factor

27 ÷ 5 = 5 remainder 2 → 5 is not a factor

27 ÷ 6 = 4 remainder 3 → 6 is not a factor

27 ÷ 7 = 3 remainder 6 → 7 is not a factor

27 ÷ 8 = 3 remainder 3 → 8 is not a factor

27 ÷ 9 = 3 remainder 0 → 9 is a factor

27 ÷ 10 = 2 remainder 7 → 10 is not a factor

27 ÷ 11 = 2 remainder 5 → 11 is not a factor

27 ÷ 12 = 2 remainder 3 → 12 is not a factor

27 ÷ 13 = 2 remainder 1 → 13 is not a factor

27 ÷ 14 = 1 remainder 13 → 14 is not a factor

27 ÷ 15 = 1 remainder 12 → 15 is not a factor

27 ÷ 16 = 1 remainder 11 → 16 is not a factor

27 ÷ 17 = 1 remainder 10 → 17 is not a factor

27 ÷ 18 = 1 remainder 9 → 18 is not a factor

27 ÷ 19 = 1 remainder 8 → 19 is not a factor

27 ÷ 20 = 1 remainder 7 → 20 is not a factor

27 ÷ 21 = 1 remainder 6 → 21 is not a factor

27 ÷ 22 = 1 remainder 5 → 22 is not a factor

27 ÷ 23 = 1 remainder 4 → 23 is not a factor

27 ÷ 24 = 1 remainder 3 → 24 is not a factor

27 ÷ 25 = 1 remainder 2 → 25 is not a factor

27 ÷ 26 = 1 remainder 1 → 26 is not a factor

27 ÷ 27 = 1 remainder 0 → 27 is a factor

Step 14: Verify 29

29 ÷ 1 = 29 remainder 0 → 1 is a factor

29 ÷ 2 = 14 remainder 1 → 2 is not a factor

29 ÷ 3 = 9 remainder 2 → 3 is not a factor

29 ÷ 4 = 7 remainder 1 → 4 is not a factor

29 ÷ 5 = 5 remainder 4 → 5 is not a factor

29 ÷ 6 = 4 remainder 5 → 6 is not a factor

29 ÷ 7 = 4 remainder 1 → 7 is not a factor

29 ÷ 8 = 3 remainder 5 → 8 is not a factor

29 ÷ 9 = 3 remainder 2 → 9 is not a factor

29 ÷ 10 = 2 remainder 9 → 10 is not a factor

29 ÷ 11 = 2 remainder 7 → 11 is not a factor

29 ÷ 12 = 2 remainder 5 → 12 is not a factor

29 ÷ 13 = 2 remainder 3 → 13 is not a factor

29 ÷ 14 = 2 remainder 1 → 14 is not a factor

29 ÷ 15 = 1 remainder 14 → 15 is not a factor

29 ÷ 16 = 1 remainder 13 → 16 is not a factor

29 ÷ 17 = 1 remainder 12 → 17 is not a factor

29 ÷ 18 = 1 remainder 11 → 18 is not a factor

29 ÷ 19 = 1 remainder 10 → 19 is not a factor

29 ÷ 20 = 1 remainder 9 → 20 is not a factor

29 ÷ 21 = 1 remainder 8 → 21 is not a factor

29 ÷ 22 = 1 remainder 7 → 22 is not a factor

29 ÷ 23 = 1 remainder 6 → 23 is not a factor

29 ÷ 24 = 1 remainder 5 → 24 is not a factor

29 ÷ 25 = 1 remainder 4 → 25 is not a factor

29 ÷ 26 = 1 remainder 3 → 26 is not a factor

29 ÷ 27 = 1 remainder 2 → 27 is not a factor

29 ÷ 28 = 1 remainder 1 → 28 is not a factor

29 ÷ 29 = 1 remainder 0 → 29 is a factor

Step 15: Verify 31

31 ÷ 1 = 31 remainder 0 → 1 is a factor

31 ÷ 2 = 15 remainder 1 → 2 is not a factor

31 ÷ 3 = 10 remainder 1 → 3 is not a factor

31 ÷ 4 = 7 remainder 3 → 4 is not a factor

31 ÷ 5 = 6 remainder 1 → 5 is not a factor

31 ÷ 6 = 5 remainder 1 → 6 is not a factor

31 ÷ 7 = 4 remainder 3 → 7 is not a factor

31 ÷ 8 = 3 remainder 7 → 8 is not a factor

31 ÷ 9 = 3 remainder 4 → 9 is not a factor

31 ÷ 10 = 3 remainder 1 → 10 is not a factor

31 ÷ 11 = 2 remainder 9 → 11 is not a factor

31 ÷ 12 = 2 remainder 7 → 12 is not a factor

31 ÷ 13 = 2 remainder 5 → 13 is not a factor

31 ÷ 14 = 2 remainder 3 → 14 is not a factor

31 ÷ 15 = 2 remainder 1 → 15 is not a factor

31 ÷ 16 = 1 remainder 15 → 16 is not a factor

31 ÷ 17 = 1 remainder 14 → 17 is not a factor

31 ÷ 18 = 1 remainder 13 → 18 is not a factor

31 ÷ 19 = 1 remainder 12 → 19 is not a factor

31 ÷ 20 = 1 remainder 11 → 20 is not a factor

31 ÷ 21 = 1 remainder 10 → 21 is not a factor

31 ÷ 22 = 1 remainder 9 → 22 is not a factor

31 ÷ 23 = 1 remainder 8 → 23 is not a factor

31 ÷ 24 = 1 remainder 7 → 24 is not a factor

31 ÷ 25 = 1 remainder 6 → 25 is not a factor

31 ÷ 26 = 1 remainder 5 → 26 is not a factor

31 ÷ 27 = 1 remainder 4 → 27 is not a factor

31 ÷ 28 = 1 remainder 3 → 28 is not a factor

31 ÷ 29 = 1 remainder 2 → 29 is not a factor

31 ÷ 30 = 1 remainder 1 → 30 is not a factor

31 ÷ 31 = 1 remainder 0 → 31 is a factor

Step 16: Verify 33

33 ÷ 1 = 33 remainder 0 → 1 is a factor

33 ÷ 2 = 16 remainder 1 → 2 is not a factor

33 ÷ 3 = 11 remainder 0 → 3 is a factor

33 ÷ 4 = 8 remainder 1 → 4 is not a factor

33 ÷ 5 = 6 remainder 3 → 5 is not a factor

33 ÷ 6 = 5 remainder 3 → 6 is not a factor

33 ÷ 7 = 4 remainder 5 → 7 is not a factor

33 ÷ 8 = 4 remainder 1 → 8 is not a factor

33 ÷ 9 = 3 remainder 6 → 9 is not a factor

33 ÷ 10 = 3 remainder 3 → 10 is not a factor

33 ÷ 11 = 3 remainder 0 → 11 is a factor

33 ÷ 12 = 2 remainder 9 → 12 is not a factor

33 ÷ 13 = 2 remainder 7 → 13 is not a factor

33 ÷ 14 = 2 remainder 5 → 14 is not a factor

33 ÷ 15 = 2 remainder 3 → 15 is not a factor

33 ÷ 16 = 2 remainder 1 → 16 is not a factor

33 ÷ 17 = 1 remainder 16 → 17 is not a factor

33 ÷ 18 = 1 remainder 15 → 18 is not a factor

33 ÷ 19 = 1 remainder 14 → 19 is not a factor

33 ÷ 20 = 1 remainder 13 → 20 is not a factor

33 ÷ 21 = 1 remainder 12 → 21 is not a factor

33 ÷ 22 = 1 remainder 11 → 22 is not a factor

33 ÷ 23 = 1 remainder 10 → 23 is not a factor

33 ÷ 24 = 1 remainder 9 → 24 is not a factor

33 ÷ 25 = 1 remainder 8 → 25 is not a factor

33 ÷ 26 = 1 remainder 7 → 26 is not a factor

33 ÷ 27 = 1 remainder 6 → 27 is not a factor

33 ÷ 28 = 1 remainder 5 → 28 is not a factor

33 ÷ 29 = 1 remainder 4 → 29 is not a factor

33 ÷ 30 = 1 remainder 3 → 30 is not a factor

33 ÷ 31 = 1 remainder 2 → 31 is not a factor

33 ÷ 32 = 1 remainder 1 → 32 is not a factor

33 ÷ 33 = 1 remainder 0 → 33 is a factor

Step 17: Verify 35

35 ÷ 1 = 35 remainder 0 → 1 is a factor

35 ÷ 2 = 17 remainder 1 → 2 is not a factor

35 ÷ 3 = 11 remainder 2 → 3 is not a factor

35 ÷ 4 = 8 remainder 3 → 4 is not a factor

35 ÷ 5 = 7 remainder 0 → 5 is a factor

35 ÷ 6 = 5 remainder 5 → 6 is not a factor

35 ÷ 7 = 5 remainder 0 → 7 is a factor

35 ÷ 8 = 4 remainder 3 → 8 is not a factor

35 ÷ 9 = 3 remainder 8 → 9 is not a factor

35 ÷ 10 = 3 remainder 5 → 10 is not a factor

35 ÷ 11 = 3 remainder 2 → 11 is not a factor

35 ÷ 12 = 2 remainder 11 → 12 is not a factor

35 ÷ 13 = 2 remainder 9 → 13 is not a factor

35 ÷ 14 = 2 remainder 7 → 14 is not a factor

35 ÷ 15 = 2 remainder 5 → 15 is not a factor

35 ÷ 16 = 2 remainder 3 → 16 is not a factor

35 ÷ 17 = 2 remainder 1 → 17 is not a factor

35 ÷ 18 = 1 remainder 17 → 18 is not a factor

35 ÷ 19 = 1 remainder 16 → 19 is not a factor

35 ÷ 20 = 1 remainder 15 → 20 is not a factor

35 ÷ 21 = 1 remainder 14 → 21 is not a factor

35 ÷ 22 = 1 remainder 13 → 22 is not a factor

35 ÷ 23 = 1 remainder 12 → 23 is not a factor

35 ÷ 24 = 1 remainder 11 → 24 is not a factor

35 ÷ 25 = 1 remainder 10 → 25 is not a factor

35 ÷ 26 = 1 remainder 9 → 26 is not a factor

35 ÷ 27 = 1 remainder 8 → 27 is not a factor

35 ÷ 28 = 1 remainder 7 → 28 is not a factor

35 ÷ 29 = 1 remainder 6 → 29 is not a factor

35 ÷ 30 = 1 remainder 5 → 30 is not a factor

35 ÷ 31 = 1 remainder 4 → 31 is not a factor

35 ÷ 32 = 1 remainder 3 → 32 is not a factor

35 ÷ 33 = 1 remainder 2 → 33 is not a factor

35 ÷ 34 = 1 remainder 1 → 34 is not a factor

35 ÷ 35 = 1 remainder 0 → 35 is a factor

Step 18: Verify 37

37 ÷ 1 = 37 remainder 0 → 1 is a factor

37 ÷ 2 = 18 remainder 1 → 2 is not a factor

37 ÷ 3 = 12 remainder 1 → 3 is not a factor

37 ÷ 4 = 9 remainder 1 → 4 is not a factor

37 ÷ 5 = 7 remainder 2 → 5 is not a factor

37 ÷ 6 = 6 remainder 1 → 6 is not a factor

37 ÷ 7 = 5 remainder 2 → 7 is not a factor

37 ÷ 8 = 4 remainder 5 → 8 is not a factor

37 ÷ 9 = 4 remainder 1 → 9 is not a factor

37 ÷ 10 = 3 remainder 7 → 10 is not a factor

37 ÷ 11 = 3 remainder 4 → 11 is not a factor

37 ÷ 12 = 3 remainder 1 → 12 is not a factor

37 ÷ 13 = 2 remainder 11 → 13 is not a factor

37 ÷ 14 = 2 remainder 9 → 14 is not a factor

37 ÷ 15 = 2 remainder 7 → 15 is not a factor

37 ÷ 16 = 2 remainder 5 → 16 is not a factor

37 ÷ 17 = 2 remainder 3 → 17 is not a factor

37 ÷ 18 = 2 remainder 1 → 18 is not a factor

37 ÷ 19 = 1 remainder 18 → 19 is not a factor

37 ÷ 20 = 1 remainder 17 → 20 is not a factor

37 ÷ 21 = 1 remainder 16 → 21 is not a factor

37 ÷ 22 = 1 remainder 15 → 22 is not a factor

37 ÷ 23 = 1 remainder 14 → 23 is not a factor

37 ÷ 24 = 1 remainder 13 → 24 is not a factor

37 ÷ 25 = 1 remainder 12 → 25 is not a factor

37 ÷ 26 = 1 remainder 11 → 26 is not a factor

37 ÷ 27 = 1 remainder 10 → 27 is not a factor

37 ÷ 28 = 1 remainder 9 → 28 is not a factor

37 ÷ 29 = 1 remainder 8 → 29 is not a factor

37 ÷ 30 = 1 remainder 7 → 30 is not a factor

37 ÷ 31 = 1 remainder 6 → 31 is not a factor

37 ÷ 32 = 1 remainder 5 → 32 is not a factor

37 ÷ 33 = 1 remainder 4 → 33 is not a factor

37 ÷ 34 = 1 remainder 3 → 34 is not a factor

37 ÷ 35 = 1 remainder 2 → 35 is not a factor

37 ÷ 36 = 1 remainder 1 → 36 is not a factor

37 ÷ 37 = 1 remainder 0 → 37 is a factor

Prime numbers: 23, 29, 31, 37 | Composite numbers: 25, 27, 33, 35
Final answer:

Prime numbers in the list: 23, 29, 31, 37

Composite numbers in the list: 25, 27, 33, 35

Applied rules:

• Prime Definition: Must have exactly two factors

• Factor Testing: Check divisibility systematically

• Counting: Only 2 factors means it's prime

2 Worksheet: Composite Identification
Worksheet 2
Underline all the composite numbers in this list: 44, 45, 46, 47, 48, 49, 50, 51
Definition:

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

Numbers
44, 45, 46, 47, 48, 49, 50, 51
Factors
List each
Types
C, C, C, P, C, C, C, C
Step 1: Analyze 44

Factors of 44: 1, 2, 4, 11, 22, 44 → Composite (6 factors)

Step 2: Analyze 45

Factors of 45: 1, 3, 5, 9, 15, 45 → Composite (6 factors)

Step 3: Analyze 46

Factors of 46: 1, 2, 23, 46 → Composite (4 factors)

Step 4: Analyze 47

Factors of 47: 1, 47 → Prime (exactly 2 factors)

Step 5: Analyze 48

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 → Composite (10 factors)

Step 6: Analyze 49

Factors of 49: 1, 7, 49 → Composite (3 factors)

Step 7: Analyze 50

Factors of 50: 1, 2, 5, 10, 25, 50 → Composite (6 factors)

Step 8: Analyze 51

Factors of 51: 1, 3, 17, 51 → Composite (4 factors)

Step 9: Underline the composites

Underline: 44, 45, 46, 48, 49, 50, 51

Step 10: Leave the primes

Leave ununderlined: 47

Step 11: Verify 44

44 ÷ 1 = 44 remainder 0 → 1 is a factor

44 ÷ 2 = 22 remainder 0 → 2 is a factor

44 ÷ 3 = 14 remainder 2 → 3 is not a factor

44 ÷ 4 = 11 remainder 0 → 4 is a factor

44 ÷ 5 = 8 remainder 4 → 5 is not a factor

44 ÷ 6 = 7 remainder 2 → 6 is not a factor

44 ÷ 7 = 6 remainder 2 → 7 is not a factor

44 ÷ 8 = 5 remainder 4 → 8 is not a factor

44 ÷ 9 = 4 remainder 8 → 9 is not a factor

44 ÷ 10 = 4 remainder 4 → 10 is not a factor

44 ÷ 11 = 4 remainder 0 → 11 is a factor

44 ÷ 12 = 3 remainder 8 → 12 is not a factor

44 ÷ 13 = 3 remainder 5 → 13 is not a factor

44 ÷ 14 = 3 remainder 2 → 14 is not a factor

44 ÷ 15 = 2 remainder 14 → 15 is not a factor

44 ÷ 16 = 2 remainder 12 → 16 is not a factor

44 ÷ 17 = 2 remainder 10 → 17 is not a factor

44 ÷ 18 = 2 remainder 8 → 18 is not a factor

44 ÷ 19 = 2 remainder 6 → 19 is not a factor

44 ÷ 20 = 2 remainder 4 → 20 is not a factor

44 ÷ 21 = 2 remainder 2 → 21 is not a factor

44 ÷ 22 = 2 remainder 0 → 22 is a factor

44 ÷ 23 = 1 remainder 21 → 23 is not a factor

44 ÷ 24 = 1 remainder 20 → 24 is not a factor

44 ÷ 25 = 1 remainder 19 → 25 is not a factor

44 ÷ 26 = 1 remainder 18 → 26 is not a factor

44 ÷ 27 = 1 remainder 17 → 27 is not a factor

44 ÷ 28 = 1 remainder 16 → 28 is not a factor

44 ÷ 29 = 1 remainder 15 → 29 is not a factor

44 ÷ 30 = 1 remainder 14 → 30 is not a factor

44 ÷ 31 = 1 remainder 13 → 31 is not a factor

44 ÷ 32 = 1 remainder 12 → 32 is not a factor

44 ÷ 33 = 1 remainder 11 → 33 is not a factor

44 ÷ 34 = 1 remainder 10 → 34 is not a factor

44 ÷ 35 = 1 remainder 9 → 35 is not a factor

44 ÷ 36 = 1 remainder 8 → 36 is not a factor

44 ÷ 37 = 1 remainder 7 → 37 is not a factor

44 ÷ 38 = 1 remainder 6 → 38 is not a factor

44 ÷ 39 = 1 remainder 5 → 39 is not a factor

44 ÷ 40 = 1 remainder 4 → 40 is not a factor

44 ÷ 41 = 1 remainder 3 → 41 is not a factor

44 ÷ 42 = 1 remainder 2 → 42 is not a factor

44 ÷ 43 = 1 remainder 1 → 43 is not a factor

44 ÷ 44 = 1 remainder 0 → 44 is a factor

Step 12: Verify 45

45 ÷ 1 = 45 remainder 0 → 1 is a factor

45 ÷ 2 = 22 remainder 1 → 2 is not a factor

45 ÷ 3 = 15 remainder 0 → 3 is a factor

45 ÷ 4 = 11 remainder 1 → 4 is not a factor

45 ÷ 5 = 9 remainder 0 → 5 is a factor

45 ÷ 6 = 7 remainder 3 → 6 is not a factor

45 ÷ 7 = 6 remainder 3 → 7 is not a factor

45 ÷ 8 = 5 remainder 5 → 8 is not a factor

45 ÷ 9 = 5 remainder 0 → 9 is a factor

45 ÷ 10 = 4 remainder 5 → 10 is not a factor

45 ÷ 11 = 4 remainder 1 → 11 is not a factor

45 ÷ 12 = 3 remainder 9 → 12 is not a factor

45 ÷ 13 = 3 remainder 6 → 13 is not a factor

45 ÷ 14 = 3 remainder 3 → 14 is not a factor

45 ÷ 15 = 3 remainder 0 → 15 is a factor

45 ÷ 16 = 2 remainder 13 → 16 is not a factor

45 ÷ 17 = 2 remainder 11 → 17 is not a factor

45 ÷ 18 = 2 remainder 9 → 18 is not a factor

45 ÷ 19 = 2 remainder 7 → 19 is not a factor

45 ÷ 20 = 2 remainder 5 → 20 is not a factor

45 ÷ 21 = 2 remainder 3 → 21 is not a factor

45 ÷ 22 = 2 remainder 1 → 22 is not a factor

45 ÷ 23 = 1 remainder 22 → 23 is not a factor

45 ÷ 24 = 1 remainder 21 → 24 is not a factor

45 ÷ 25 = 1 remainder 20 → 25 is not a factor

45 ÷ 26 = 1 remainder 19 → 26 is not a factor

45 ÷ 27 = 1 remainder 18 → 27 is not a factor

45 ÷ 28 = 1 remainder 17 → 28 is not a factor

45 ÷ 29 = 1 remainder 16 → 29 is not a factor

45 ÷ 30 = 1 remainder 15 → 30 is not a factor

45 ÷ 31 = 1 remainder 14 → 31 is not a factor

45 ÷ 32 = 1 remainder 13 → 32 is not a factor

45 ÷ 33 = 1 remainder 12 → 33 is not a factor

45 ÷ 34 = 1 remainder 11 → 34 is not a factor

45 ÷ 35 = 1 remainder 10 → 35 is not a factor

45 ÷ 36 = 1 remainder 9 → 36 is not a factor

45 ÷ 37 = 1 remainder 8 → 37 is not a factor

45 ÷ 38 = 1 remainder 7 → 38 is not a factor

45 ÷ 39 = 1 remainder 6 → 39 is not a factor

45 ÷ 40 = 1 remainder 5 → 40 is not a factor

45 ÷ 41 = 1 remainder 4 → 41 is not a factor

45 ÷ 42 = 1 remainder 3 → 42 is not a factor

45 ÷ 43 = 1 remainder 2 → 43 is not a factor

45 ÷ 44 = 1 remainder 1 → 44 is not a factor

45 ÷ 45 = 1 remainder 0 → 45 is a factor

Composite numbers: 44, 45, 46, 48, 49, 50, 51 | Prime: 47
Final answer:

Composite numbers in the list: 44, 45, 46, 48, 49, 50, 51

Prime number in the list: 47

Applied rules:

• Composite Definition: Must have more than two factors

• Factor Testing: Check divisibility systematically

• Counting: More than 2 factors means it's composite

3 Worksheet: Mixed Classification
Worksheet 3
From the numbers below, write "P" next to primes and "C" next to composites: 53, 54, 55, 56, 57, 58, 59, 60
Definition:

Mixed classification: Determining if each number is prime or composite by finding its factors

Numbers
53, 54, 55, 56, 57, 58, 59, 60
Factors
List each
Types
P, C, C, C, C, C, P, C
Step 1: Analyze 53

Factors: 1, 53 → Prime (exactly 2 factors)

Step 2: Analyze 54

Factors: 1, 2, 3, 6, 9, 18, 27, 54 → Composite (8 factors)

Step 3: Analyze 55

Factors: 1, 5, 11, 55 → Composite (4 factors)

Step 4: Analyze 56

Factors: 1, 2, 4, 7, 8, 14, 28, 56 → Composite (8 factors)

Step 5: Analyze 57

Factors: 1, 3, 19, 57 → Composite (4 factors)

Step 6: Analyze 58

Factors: 1, 2, 29, 58 → Composite (4 factors)

Step 7: Analyze 59

Factors: 1, 59 → Prime (exactly 2 factors)

Step 8: Analyze 60

Factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 → Composite (12 factors)

Step 9: Final classification

53-P, 54-C, 55-C, 56-C, 57-C, 58-C, 59-P, 60-C

Step 10: Verification

Primes: 53, 59 | Composites: 54, 55, 56, 57, 58, 60

Primes: 53, 59 | Composites: 54, 55, 56, 57, 58, 60
Final answer:

Classification: 53-P, 54-C, 55-C, 56-C, 57-C, 58-C, 59-P, 60-C

Applied rules:

• Systematic Analysis: Examine each number individually

• Factor Counting: Determine the exact number of factors

• Classification Rules: 2 factors = prime, more than 2 = composite

Test Concepts & Methods
Prime: 2 Factors | Composite: >2 Factors
Classification Rule
Rule 1
Prime: exactly 2 factors
Prime Definition
Rule 2
Composite: more than 2 factors
Composite Definition
Rule 3
Special: 1 = neither prime nor composite
Special Case
Key definitions:

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

Test methodology:
  1. Check if number > 1: Prime/composite only applies to numbers greater than 1
  2. Find all factors: Systematically check divisibility from 1 up to the number
  3. Count factors: Tally the total number of factors found
  4. Classify: 2 factors = prime, more than 2 = composite
Tip 1: Even numbers (except 2) are always composite since they're divisible by 2.
Tip 2: Numbers ending in 5 (except 5) are composite since they're divisible by 5.
Tip 3: The number 2 is the only even prime number.
Tip 4: To save time, only test divisors up to the square root of the number.
Common errors: Thinking 1 is prime, missing factors when counting, confusing factors with multiples.
Key concepts: Prime numbers are the building blocks of all numbers, every composite number can be broken down into primes.
Rules for tests:

• 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, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109...

• Every composite number has at least one prime factor

Test Solutions: 4 to 5
4 Test: Word Problem
Test Problem 4
A teacher has 67 books. She wants to arrange them in equal stacks with no books left over. How many different ways can she stack the books? Is 67 prime or composite?
Definition:

Word problems: Real-world situations requiring knowledge of factors and divisibility

Book count
67
Factors
1, 67
Stacking options
1×67 or 67×1
Step 1: Find factors of 67

Factors of 67: 1, 67

Step 2: Interpret stacking options

Option 1: 1 stack of 67 books (1×67)

Option 2: 67 stacks of 1 book each (67×1)

Step 3: Determine if 67 is prime or composite

Since 67 has exactly 2 factors (1 and 67), it's prime

Step 4: Answer the question

The teacher can stack her books in only 2 ways: 1 stack of 67 books or 67 stacks of 1 book each

Step 5: Verify 67

67 ÷ 1 = 67 remainder 0 → 1 is a factor

67 ÷ 2 = 33 remainder 1 → 2 is not a factor

67 ÷ 3 = 22 remainder 1 → 3 is not a factor

67 ÷ 4 = 16 remainder 3 → 4 is not a factor

67 ÷ 5 = 13 remainder 2 → 5 is not a factor

67 ÷ 6 = 11 remainder 1 → 6 is not a factor

67 ÷ 7 = 9 remainder 4 → 7 is not a factor

67 ÷ 8 = 8 remainder 3 → 8 is not a factor

67 ÷ 9 = 7 remainder 4 → 9 is not a factor

67 ÷ 10 = 6 remainder 7 → 10 is not a factor

67 ÷ 11 = 6 remainder 1 → 11 is not a factor

67 ÷ 12 = 5 remainder 7 → 12 is not a factor

67 ÷ 13 = 5 remainder 2 → 13 is not a factor

67 ÷ 14 = 4 remainder 11 → 14 is not a factor

67 ÷ 15 = 4 remainder 7 → 15 is not a factor

67 ÷ 16 = 4 remainder 3 → 16 is not a factor

67 ÷ 17 = 3 remainder 16 → 17 is not a factor

67 ÷ 18 = 3 remainder 13 → 18 is not a factor

67 ÷ 19 = 3 remainder 10 → 19 is not a factor

67 ÷ 20 = 3 remainder 7 → 20 is not a factor

67 ÷ 21 = 3 remainder 4 → 21 is not a factor

67 ÷ 22 = 3 remainder 1 → 22 is not a factor

67 ÷ 23 = 2 remainder 21 → 23 is not a factor

67 ÷ 24 = 2 remainder 19 → 24 is not a factor

67 ÷ 25 = 2 remainder 17 → 25 is not a factor

67 ÷ 26 = 2 remainder 15 → 26 is not a factor

67 ÷ 27 = 2 remainder 13 → 27 is not a factor

67 ÷ 28 = 2 remainder 11 → 28 is not a factor

67 ÷ 29 = 2 remainder 9 → 29 is not a factor

67 ÷ 30 = 2 remainder 7 → 30 is not a factor

67 ÷ 31 = 2 remainder 5 → 31 is not a factor

67 ÷ 32 = 2 remainder 3 → 32 is not a factor

67 ÷ 33 = 2 remainder 1 → 33 is not a factor

67 ÷ 34 = 1 remainder 33 → 34 is not a factor

67 ÷ 35 = 1 remainder 32 → 35 is not a factor

67 ÷ 36 = 1 remainder 31 → 36 is not a factor

67 ÷ 37 = 1 remainder 30 → 37 is not a factor

67 ÷ 38 = 1 remainder 29 → 38 is not a factor

67 ÷ 39 = 1 remainder 28 → 39 is not a factor

67 ÷ 40 = 1 remainder 27 → 40 is not a factor

67 ÷ 41 = 1 remainder 26 → 41 is not a factor

67 ÷ 42 = 1 remainder 25 → 42 is not a factor

67 ÷ 43 = 1 remainder 24 → 43 is not a factor

67 ÷ 44 = 1 remainder 23 → 44 is not a factor

67 ÷ 45 = 1 remainder 22 → 45 is not a factor

67 ÷ 46 = 1 remainder 21 → 46 is not a factor

67 ÷ 47 = 1 remainder 20 → 47 is not a factor

67 ÷ 48 = 1 remainder 19 → 48 is not a factor

67 ÷ 49 = 1 remainder 18 → 49 is not a factor

67 ÷ 50 = 1 remainder 17 → 50 is not a factor

67 ÷ 51 = 1 remainder 16 → 51 is not a factor

67 ÷ 52 = 1 remainder 15 → 52 is not a factor

67 ÷ 53 = 1 remainder 14 → 53 is not a factor

67 ÷ 54 = 1 remainder 13 → 54 is not a factor

67 ÷ 55 = 1 remainder 12 → 55 is not a factor

67 ÷ 56 = 1 remainder 11 → 56 is not a factor

67 ÷ 57 = 1 remainder 10 → 57 is not a factor

67 ÷ 58 = 1 remainder 9 → 58 is not a factor

67 ÷ 59 = 1 remainder 8 → 59 is not a factor

67 ÷ 60 = 1 remainder 7 → 60 is not a factor

67 ÷ 61 = 1 remainder 6 → 61 is not a factor

67 ÷ 62 = 1 remainder 5 → 62 is not a factor

67 ÷ 63 = 1 remainder 4 → 63 is not a factor

67 ÷ 64 = 1 remainder 3 → 64 is not a factor

67 ÷ 65 = 1 remainder 2 → 65 is not a factor

67 ÷ 66 = 1 remainder 1 → 66 is not a factor

67 ÷ 67 = 1 remainder 0 → 67 is a factor

67 is prime, so teacher has only 2 stacking options
Final answer:

The teacher can stack her 67 books in 1 stack of 67 books or 67 stacks of 1 book each. The number 67 is prime because it has exactly two factors: 1 and 67.

Applied rules:

• Prime Application: Prime numbers have limited arrangement options

• Real-world connection: Stacking items requires understanding factors

• Verification: Confirm that arrangements use all items

5 Test: Pattern Recognition
Test Problem 5
Find all prime numbers between 70 and 80. List them and explain your reasoning.
Definition:

Range analysis: Systematically checking numbers within a specified range

Range
70-80
Check each
71, 72, 73, 74, 75, 76, 77, 78, 79
Primes
71, 73, 79
Step 1: List numbers 71-79 (excluding 70 and 80)

71, 72, 73, 74, 75, 76, 77, 78, 79

Step 2: Check 71

Factors: 1, 71 → Prime

Step 3: Check 72

72 = 8 × 9 → Composite

Step 4: Check 73

Factors: 1, 73 → Prime

Step 5: Check 74

74 = 2 × 37 → Composite

Step 6: Check 75

75 = 3 × 25 → Composite

Step 7: Check 76

76 = 4 × 19 → Composite

Step 8: Check 77

77 = 7 × 11 → Composite

Step 9: Check 78

78 = 6 × 13 → Composite

Step 10: Check 79

Factors: 1, 79 → Prime

Prime numbers between 70 and 80: 71, 73, 79
Final answer:

The prime numbers between 70 and 80 are: 71, 73, and 79

Applied rules:

• Systematic approach: Check each number in the range individually

• Even number rule: All even numbers > 2 are composite

• Verification: Double-check each classification

Test Concepts and Key Ideas
Prime: 2 Factors | Composite: >2 Factors
Test Classification
Test definitions:

Prime Number Tests: Assessments focusing on identifying numbers with exactly two factors

Composite Number Tests: Assessments focusing on identifying numbers with more than two factors

Classification Tests: Evaluations that require distinguishing between prime and composite numbers

Test approach:
  1. Test 1 (Prime): Take number 79 → factors are 1 and 79 → exactly 2 factors → prime
  2. Test 2 (Composite): Take number 80 → factors are 1, 2, 4, 5, 8, 10, 16, 20, 40, 80 → 10 factors → composite
  3. Test 3 (Pattern): Notice that all even numbers > 2 are composite
  4. Test 4 (Application): Prime numbers limit how items can be arranged equally
Test Tip 1: The first few primes 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, 101, 103, 107, 109...

Test Tip 2: All primes > 2 are odd numbers (since even numbers > 2 are divisible by 2)

Test Tip 3: Primes become less frequent as numbers get larger

Test Tip 4: Every composite number can be broken down into prime factors

Test Properties: Prime numbers are like building blocks - everything else is made from them. Composite numbers are like "group projects" made up of smaller prime numbers.

Real-world Test: Computer security uses large prime numbers to keep passwords and messages safe.

Test rules:

• 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

Questions & Answers

Question: Why is 1 not a prime number? It seems like it should be since it only divides by itself.

Answer: This is a great question that confuses many students!

The definition of a prime number specifically states it must have exactly two different factors: 1 and itself. Since 1 only has one factor (itself), it doesn't meet this requirement.

Think of it like this: prime numbers are special because they can only be arranged in exactly two rectangular ways (1 row or 1 column). The number 1 can only be arranged in one way, so it gets its own special category.

So 1 is neither prime nor composite - it's just special by itself!

Question: How can I quickly tell if a number is prime without checking every single number?

Answer: You can use several smart strategies to make this easier!

  • Check divisibility by small primes first: 2, 3, 5, 7, 11
  • Only check up to the square root: If n is your number, only test divisors up to √n
  • Use divisibility rules: If the sum of digits is divisible by 3, the number is divisible by 3

For example, to check if 109 is prime:

  • It's not even, so not divisible by 2
  • Sum of digits: 1+0+9=10, not divisible by 3
  • Doesn't end in 0 or 5, so not divisible by 5
  • √109 ≈ 10.4, so we only check up to 10
  • Check 7: 109 ÷ 7 = 15 remainder 4 → not divisible
  • Check 10: 109 ÷ 10 = 10 remainder 9 → not divisible

Since we've checked all primes up to √109 and none divide 109, it's prime!

Question: Are there any patterns to help me remember which numbers are prime?

Answer: Yes, there are helpful patterns to recognize prime numbers!

  • All primes > 2 are odd (except 2, which is the only even prime)
  • Primes > 5 end in 1, 3, 7, or 9 (never 0, 2, 4, 5, 6, 8)
  • Primes become less frequent as numbers get larger

The first 25 primes to remember: 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, 101, 103, 107, 109...

Notice that after 2 and 5, all primes end in 1, 3, 7, or 9!

Question: What's the biggest prime number I need to know for grade 4?

Answer: For grade 4, focus on understanding prime numbers up to about 100, especially the smaller ones!

The most important prime numbers to know 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, 101, 103, 107, 109...

These are the prime numbers you'll encounter most often in problems. As you advance in math, you'll learn about larger primes, but these foundational ones are crucial for understanding factors, multiples, and divisibility.