Word problems: Real-world situations requiring knowledge of factors and divisibility
- Find all factors of the number (arrangement options)
- Count the total number of factors
- Determine if it's prime (2 factors) or composite (more than 2)
- Answer the question about arrangement options
Factors of 73: 1, 73
Option 1: 1 row of 73 marbles (1×73)
Option 2: 73 rows of 1 marble each (73×1)
Since 73 has exactly 2 factors (1 and 73), it's prime
Emma can arrange her marbles in only 2 ways: 1 row of 73 marbles or 73 rows of 1 marble each
73 ÷ 1 = 73 remainder 0 → 1 is a factor
73 ÷ 2 = 36 remainder 1 → 2 is not a factor
73 ÷ 3 = 24 remainder 1 → 3 is not a factor
73 ÷ 4 = 18 remainder 1 → 4 is not a factor
73 ÷ 5 = 14 remainder 3 → 5 is not a factor
73 ÷ 6 = 12 remainder 1 → 6 is not a factor
73 ÷ 7 = 10 remainder 3 → 7 is not a factor
73 ÷ 8 = 9 remainder 1 → 8 is not a factor
73 ÷ 9 = 8 remainder 1 → 9 is not a factor
73 ÷ 10 = 7 remainder 3 → 10 is not a factor
73 ÷ 11 = 6 remainder 7 → 11 is not a factor
73 ÷ 12 = 6 remainder 1 → 12 is not a factor
73 ÷ 13 = 5 remainder 8 → 13 is not a factor
73 ÷ 14 = 5 remainder 3 → 14 is not a factor
73 ÷ 15 = 4 remainder 13 → 15 is not a factor
73 ÷ 16 = 4 remainder 9 → 16 is not a factor
73 ÷ 17 = 4 remainder 5 → 17 is not a factor
73 ÷ 18 = 4 remainder 1 → 18 is not a factor
73 ÷ 19 = 3 remainder 16 → 19 is not a factor
73 ÷ 20 = 3 remainder 13 → 20 is not a factor
73 ÷ 21 = 3 remainder 10 → 21 is not a factor
73 ÷ 22 = 3 remainder 7 → 22 is not a factor
73 ÷ 23 = 3 remainder 4 → 23 is not a factor
73 ÷ 24 = 3 remainder 1 → 24 is not a factor
73 ÷ 25 = 2 remainder 23 → 25 is not a factor
73 ÷ 26 = 2 remainder 21 → 26 is not a factor
73 ÷ 27 = 2 remainder 19 → 27 is not a factor
73 ÷ 28 = 2 remainder 17 → 28 is not a factor
73 ÷ 29 = 2 remainder 15 → 29 is not a factor
73 ÷ 30 = 2 remainder 13 → 30 is not a factor
73 ÷ 31 = 2 remainder 11 → 31 is not a factor
73 ÷ 32 = 2 remainder 9 → 32 is not a factor
73 ÷ 33 = 2 remainder 7 → 33 is not a factor
73 ÷ 34 = 2 remainder 5 → 34 is not a factor
73 ÷ 35 = 2 remainder 3 → 35 is not a factor
73 ÷ 36 = 2 remainder 1 → 36 is not a factor
73 ÷ 37 = 1 remainder 36 → 37 is not a factor
73 ÷ 38 = 1 remainder 35 → 38 is not a factor
73 ÷ 39 = 1 remainder 34 → 39 is not a factor
73 ÷ 40 = 1 remainder 33 → 40 is not a factor
73 ÷ 41 = 1 remainder 32 → 41 is not a factor
73 ÷ 42 = 1 remainder 31 → 42 is not a factor
73 ÷ 43 = 1 remainder 30 → 43 is not a factor
73 ÷ 44 = 1 remainder 29 → 44 is not a factor
73 ÷ 45 = 1 remainder 28 → 45 is not a factor
73 ÷ 46 = 1 remainder 27 → 46 is not a factor
73 ÷ 47 = 1 remainder 26 → 47 is not a factor
73 ÷ 48 = 1 remainder 25 → 48 is not a factor
73 ÷ 49 = 1 remainder 24 → 49 is not a factor
73 ÷ 50 = 1 remainder 23 → 50 is not a factor
73 ÷ 51 = 1 remainder 22 → 51 is not a factor
73 ÷ 52 = 1 remainder 21 → 52 is not a factor
73 ÷ 53 = 1 remainder 20 → 53 is not a factor
73 ÷ 54 = 1 remainder 19 → 54 is not a factor
73 ÷ 55 = 1 remainder 18 → 55 is not a factor
73 ÷ 56 = 1 remainder 17 → 56 is not a factor
73 ÷ 57 = 1 remainder 16 → 57 is not a factor
73 ÷ 58 = 1 remainder 15 → 58 is not a factor
73 ÷ 59 = 1 remainder 14 → 59 is not a factor
73 ÷ 60 = 1 remainder 13 → 60 is not a factor
73 ÷ 61 = 1 remainder 12 → 61 is not a factor
73 ÷ 62 = 1 remainder 11 → 62 is not a factor
73 ÷ 63 = 1 remainder 10 → 63 is not a factor
73 ÷ 64 = 1 remainder 9 → 64 is not a factor
73 ÷ 65 = 1 remainder 8 → 65 is not a factor
73 ÷ 66 = 1 remainder 7 → 66 is not a factor
73 ÷ 67 = 1 remainder 6 → 67 is not a factor
73 ÷ 68 = 1 remainder 5 → 68 is not a factor
73 ÷ 69 = 1 remainder 4 → 69 is not a factor
73 ÷ 70 = 1 remainder 3 → 70 is not a factor
73 ÷ 71 = 1 remainder 2 → 71 is not a factor
73 ÷ 72 = 1 remainder 1 → 72 is not a factor
73 ÷ 73 = 1 remainder 0 → 73 is a factor
Emma can arrange her 73 marbles in 1 row of 73 marbles or 73 rows of 1 marble each. The number 73 is prime because it has exactly two factors: 1 and 73.
• Prime Application: Prime numbers limit arrangement options
• Real-world connection: Equal arrangement requires understanding factors
• Verification: Confirm that arrangements use all items
Word problems: Real-world situations requiring knowledge of divisibility and grouping
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
1 group of 60 students (1×60)
2 groups of 30 students (2×30)
3 groups of 20 students (3×20)
4 groups of 15 students (4×15)
5 groups of 12 students (5×12)
6 groups of 10 students (6×10)
10 groups of 6 students (10×6)
12 groups of 5 students (12×5)
15 groups of 4 students (15×4)
20 groups of 3 students (20×3)
30 groups of 2 students (30×2)
60 groups of 1 student (60×1)
Since 60 has 12 factors (>2), it's composite
The teacher has 12 different ways to divide 60 students into equal groups
60 ÷ 1 = 60 remainder 0 → 1 is a factor
60 ÷ 2 = 30 remainder 0 → 2 is a factor
60 ÷ 3 = 20 remainder 0 → 3 is a factor
60 ÷ 4 = 15 remainder 0 → 4 is a factor
60 ÷ 5 = 12 remainder 0 → 5 is a factor
60 ÷ 6 = 10 remainder 0 → 6 is a factor
60 ÷ 7 = 8 remainder 4 → 7 is not a factor
60 ÷ 8 = 7 remainder 4 → 8 is not a factor
60 ÷ 9 = 6 remainder 6 → 9 is not a factor
60 ÷ 10 = 6 remainder 0 → 10 is a factor
60 ÷ 11 = 5 remainder 5 → 11 is not a factor
60 ÷ 12 = 5 remainder 0 → 12 is a factor
60 ÷ 13 = 4 remainder 8 → 13 is not a factor
60 ÷ 14 = 4 remainder 4 → 14 is not a factor
60 ÷ 15 = 4 remainder 0 → 15 is a factor
60 ÷ 16 = 3 remainder 12 → 16 is not a factor
60 ÷ 17 = 3 remainder 9 → 17 is not a factor
60 ÷ 18 = 3 remainder 6 → 18 is not a factor
60 ÷ 19 = 3 remainder 3 → 19 is not a factor
60 ÷ 20 = 3 remainder 0 → 20 is a factor
60 ÷ 21 = 2 remainder 18 → 21 is not a factor
60 ÷ 22 = 2 remainder 16 → 22 is not a factor
60 ÷ 23 = 2 remainder 14 → 23 is not a factor
60 ÷ 24 = 2 remainder 12 → 24 is not a factor
60 ÷ 25 = 2 remainder 10 → 25 is not a factor
60 ÷ 26 = 2 remainder 8 → 26 is not a factor
60 ÷ 27 = 2 remainder 6 → 27 is not a factor
60 ÷ 28 = 2 remainder 4 → 28 is not a factor
60 ÷ 29 = 2 remainder 2 → 29 is not a factor
60 ÷ 30 = 2 remainder 0 → 30 is a factor
60 ÷ 31 = 1 remainder 29 → 31 is not a factor
60 ÷ 32 = 1 remainder 28 → 32 is not a factor
60 ÷ 33 = 1 remainder 27 → 33 is not a factor
60 ÷ 34 = 1 remainder 26 → 34 is not a factor
60 ÷ 35 = 1 remainder 25 → 35 is not a factor
60 ÷ 36 = 1 remainder 24 → 36 is not a factor
60 ÷ 37 = 1 remainder 23 → 37 is not a factor
60 ÷ 38 = 1 remainder 22 → 38 is not a factor
60 ÷ 39 = 1 remainder 21 → 39 is not a factor
60 ÷ 40 = 1 remainder 20 → 40 is not a factor
60 ÷ 41 = 1 remainder 19 → 41 is not a factor
60 ÷ 42 = 1 remainder 18 → 42 is not a factor
60 ÷ 43 = 1 remainder 17 → 43 is not a factor
60 ÷ 44 = 1 remainder 16 → 44 is not a factor
60 ÷ 45 = 1 remainder 15 → 45 is not a factor
60 ÷ 46 = 1 remainder 14 → 46 is not a factor
60 ÷ 47 = 1 remainder 13 → 47 is not a factor
60 ÷ 48 = 1 remainder 12 → 48 is not a factor
60 ÷ 49 = 1 remainder 11 → 49 is not a factor
60 ÷ 50 = 1 remainder 10 → 50 is not a factor
60 ÷ 51 = 1 remainder 9 → 51 is not a factor
60 ÷ 52 = 1 remainder 8 → 52 is not a factor
60 ÷ 53 = 1 remainder 7 → 53 is not a factor
60 ÷ 54 = 1 remainder 6 → 54 is not a factor
60 ÷ 55 = 1 remainder 5 → 55 is not a factor
60 ÷ 56 = 1 remainder 4 → 56 is not a factor
60 ÷ 57 = 1 remainder 3 → 57 is not a factor
60 ÷ 58 = 1 remainder 2 → 58 is not a factor
60 ÷ 59 = 1 remainder 1 → 59 is not a factor
60 ÷ 60 = 1 remainder 0 → 60 is a factor
The teacher can divide her 60 students into equal groups in 12 different ways. The number 60 is composite because it has more than two factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
• Composite Application: Composite numbers offer many grouping options
• Real-world connection: Dividing students requires understanding factors
• Verification: Confirm that groupings use all students
Word problems: Real-world scenarios involving factors and divisibility patterns
Factors of 97: 1, 97
Option 1: 1 row of 97 trees (1×97)
Option 2: 97 rows of 1 tree each (97×1)
Since 97 has exactly 2 factors (1 and 97), it's prime
The farmer can arrange his trees in only 2 ways: 1 row of 97 trees or 97 rows of 1 tree each
97 ÷ 1 = 97 remainder 0 → 1 is a factor
97 ÷ 2 = 48 remainder 1 → 2 is not a factor
97 ÷ 3 = 32 remainder 1 → 3 is not a factor
97 ÷ 4 = 24 remainder 1 → 4 is not a factor
97 ÷ 5 = 19 remainder 2 → 5 is not a factor
97 ÷ 6 = 16 remainder 1 → 6 is not a factor
97 ÷ 7 = 13 remainder 6 → 7 is not a factor
97 ÷ 8 = 12 remainder 1 → 8 is not a factor
97 ÷ 9 = 10 remainder 7 → 9 is not a factor
97 ÷ 10 = 9 remainder 7 → 10 is not a factor
97 ÷ 11 = 8 remainder 9 → 11 is not a factor
97 ÷ 12 = 8 remainder 1 → 12 is not a factor
97 ÷ 13 = 7 remainder 6 → 13 is not a factor
97 ÷ 14 = 6 remainder 13 → 14 is not a factor
97 ÷ 15 = 6 remainder 7 → 15 is not a factor
97 ÷ 16 = 6 remainder 1 → 16 is not a factor
97 ÷ 17 = 5 remainder 12 → 17 is not a factor
97 ÷ 18 = 5 remainder 7 → 18 is not a factor
97 ÷ 19 = 5 remainder 2 → 19 is not a factor
97 ÷ 20 = 4 remainder 17 → 20 is not a factor
97 ÷ 21 = 4 remainder 13 → 21 is not a factor
97 ÷ 22 = 4 remainder 9 → 22 is not a factor
97 ÷ 23 = 4 remainder 5 → 23 is not a factor
97 ÷ 24 = 4 remainder 1 → 24 is not a factor
97 ÷ 25 = 3 remainder 22 → 25 is not a factor
97 ÷ 26 = 3 remainder 19 → 26 is not a factor
97 ÷ 27 = 3 remainder 16 → 27 is not a factor
97 ÷ 28 = 3 remainder 13 → 28 is not a factor
97 ÷ 29 = 3 remainder 10 → 29 is not a factor
97 ÷ 30 = 3 remainder 7 → 30 is not a factor
97 ÷ 31 = 3 remainder 4 → 31 is not a factor
97 ÷ 32 = 3 remainder 1 → 32 is not a factor
97 ÷ 33 = 2 remainder 31 → 33 is not a factor
97 ÷ 34 = 2 remainder 29 → 34 is not a factor
97 ÷ 35 = 2 remainder 27 → 35 is not a factor
97 ÷ 36 = 2 remainder 25 → 36 is not a factor
97 ÷ 37 = 2 remainder 23 → 37 is not a factor
97 ÷ 38 = 2 remainder 21 → 38 is not a factor
97 ÷ 39 = 2 remainder 19 → 39 is not a factor
97 ÷ 40 = 2 remainder 17 → 40 is not a factor
97 ÷ 41 = 2 remainder 15 → 41 is not a factor
97 ÷ 42 = 2 remainder 13 → 42 is not a factor
97 ÷ 43 = 2 remainder 11 → 43 is not a factor
97 ÷ 44 = 2 remainder 9 → 44 is not a factor
97 ÷ 45 = 2 remainder 7 → 45 is not a factor
97 ÷ 46 = 2 remainder 5 → 46 is not a factor
97 ÷ 47 = 2 remainder 3 → 47 is not a factor
97 ÷ 48 = 2 remainder 1 → 48 is not a factor
97 ÷ 49 = 1 remainder 48 → 49 is not a factor
97 ÷ 50 = 1 remainder 47 → 50 is not a factor
97 ÷ 51 = 1 remainder 46 → 51 is not a factor
97 ÷ 52 = 1 remainder 45 → 52 is not a factor
97 ÷ 53 = 1 remainder 44 → 53 is not a factor
97 ÷ 54 = 1 remainder 43 → 54 is not a factor
97 ÷ 55 = 1 remainder 42 → 55 is not a factor
97 ÷ 56 = 1 remainder 41 → 56 is not a factor
97 ÷ 57 = 1 remainder 40 → 57 is not a factor
97 ÷ 58 = 1 remainder 39 → 58 is not a factor
97 ÷ 59 = 1 remainder 38 → 59 is not a factor
97 ÷ 60 = 1 remainder 37 → 60 is not a factor
97 ÷ 61 = 1 remainder 36 → 61 is not a factor
97 ÷ 62 = 1 remainder 35 → 62 is not a factor
97 ÷ 63 = 1 remainder 34 → 63 is not a factor
97 ÷ 64 = 1 remainder 33 → 64 is not a factor
97 ÷ 65 = 1 remainder 32 → 65 is not a factor
97 ÷ 66 = 1 remainder 31 → 66 is not a factor
97 ÷ 67 = 1 remainder 30 → 67 is not a factor
97 ÷ 68 = 1 remainder 29 → 68 is not a factor
97 ÷ 69 = 1 remainder 28 → 69 is not a factor
97 ÷ 70 = 1 remainder 27 → 70 is not a factor
97 ÷ 71 = 1 remainder 26 → 71 is not a factor
97 ÷ 72 = 1 remainder 25 → 72 is not a factor
97 ÷ 73 = 1 remainder 24 → 73 is not a factor
97 ÷ 74 = 1 remainder 23 → 74 is not a factor
97 ÷ 75 = 1 remainder 22 → 75 is not a factor
97 ÷ 76 = 1 remainder 21 → 76 is not a factor
97 ÷ 77 = 1 remainder 20 → 77 is not a factor
97 ÷ 78 = 1 remainder 19 → 78 is not a factor
97 ÷ 79 = 1 remainder 18 → 79 is not a factor
97 ÷ 80 = 1 remainder 17 → 80 is not a factor
97 ÷ 81 = 1 remainder 16 → 81 is not a factor
97 ÷ 82 = 1 remainder 15 → 82 is not a factor
97 ÷ 83 = 1 remainder 14 → 83 is not a factor
97 ÷ 84 = 1 remainder 13 → 84 is not a factor
97 ÷ 85 = 1 remainder 12 → 85 is not a factor
97 ÷ 86 = 1 remainder 11 → 86 is not a factor
97 ÷ 87 = 1 remainder 10 → 87 is not a factor
97 ÷ 88 = 1 remainder 9 → 88 is not a factor
97 ÷ 89 = 1 remainder 8 → 89 is not a factor
97 ÷ 90 = 1 remainder 7 → 90 is not a factor
97 ÷ 91 = 1 remainder 6 → 91 is not a factor
97 ÷ 92 = 1 remainder 5 → 92 is not a factor
97 ÷ 93 = 1 remainder 4 → 93 is not a factor
97 ÷ 94 = 1 remainder 3 → 94 is not a factor
97 ÷ 95 = 1 remainder 2 → 95 is not a factor
97 ÷ 96 = 1 remainder 1 → 96 is not a factor
97 ÷ 97 = 1 remainder 0 → 97 is a factor
The farmer can arrange his 97 apple trees in 1 row of 97 trees or 97 rows of 1 tree each. The number 97 is prime because it has exactly two factors: 1 and 97.
• Prime Application: Prime numbers limit arrangement options
• Real-world connection: Planting arrangements require understanding factors
• Verification: Confirm that arrangements use all trees
Prime Number Explanation: Numbers that have exactly two factors: 1 and itself, like building blocks that can't be broken down further
Composite Number Explanation: Numbers that have more than two factors, like group projects made up of smaller numbers
Factor Explanation: Numbers that divide evenly into another number without leaving a remainder
- Explanation 1 (Prime): Take number 13 → factors are 1 and 13 → exactly 2 factors → prime
- Explanation 2 (Composite): Take number 15 → factors are 1, 3, 5, 15 → 4 factors → composite
- Explanation 3 (Pattern): Notice that all even numbers > 2 are composite
- Explanation 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