Mathpower 2 1 Factors and Divisibility | PDF - Free Printable

Let's solve each problem step by step from the worksheet titled "2.1 Factors and Divisibility".

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Section 1: State the missing factor

We are given multiplication equations and need to find the missing number.

1. $ 5 \times \_ = 45 $ → $ 45 ÷ 5 = 9 $ → 9
2. $ \_ \times 3 = 21 $ → $ 21 ÷ 3 = 7 $ → 7
3. $ 11 \times \_ = 77 $ → $ 77 ÷ 11 = 7 $ → 7
4. $ \_ \times 6 = 54 $ → $ 54 ÷ 6 = 9 $ → 9
5. $ \_ \times 7 = 49 $ → $ 49 ÷ 7 = 7 $ → 7
6. $ 2 \times \_ = 16 $ → $ 16 ÷ 2 = 8 $ → 8
7. $ \_ \times 4 = 28 $ → $ 28 ÷ 4 = 7 $ → 7
8. $ 9 \times \_ = 36 $ → $ 36 ÷ 9 = 4 $ → 4

✔ Answers:
1. 9
2. 7
3. 7
4. 9
5. 7
6. 8
7. 7
8. 4

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Section 2: Write two pairs of factors for each number

We need to write two different factor pairs (a × b) that equal the number.

9. $ \_ \times \_ = 24 $; $ \_ \times \_ = 24 $
→ $ 6 \times 4 = 24 $, $ 8 \times 3 = 24 $

10. $ \_ \times \_ = 26 $; $ \_ \times \_ = 26 $
→ $ 13 \times 2 = 26 $, $ 26 \times 1 = 26 $

11. $ \_ \times \_ = 48 $; $ \_ \times \_ = 48 $
→ $ 6 \times 8 = 48 $, $ 12 \times 4 = 48 $

12. $ \_ \times \_ = 80 $; $ \_ \times \_ = 80 $
→ $ 8 \times 10 = 80 $, $ 5 \times 16 = 80 $

13. $ \_ \times \_ = 108 $; $ \_ \times \_ = 108 $
→ $ 9 \times 12 = 108 $, $ 6 \times 18 = 108 $

✔ Answers:
9. 6 × 4, 8 × 3
10. 13 × 2, 26 × 1
11. 6 × 8, 12 × 4
12. 8 × 10, 5 × 16
13. 9 × 12, 6 × 18

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Section 3: Complete each list of factors

We are given some factors and need to fill in the blanks.

14. 32: 1, 2, __, 8, __, 32
Factors of 32: 1, 2, 4, 8, 16, 32
→ Missing: 4, 16

15. 65: 1, __, 13, __
Factors of 65: 1, 5, 13, 65
→ Missing: 5, 65

16. 80: __, 2, __, 5, __, 10, __, 20, __, 80
List all factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
So:
→ 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
Missing: 1, 4, 8, 16, 40

17. 72: __, __, 3, 4, __, __, 9, 12, 18, __, __, 72
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
So:
→ 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Missing: 1, 2, 6, 8, 24, 36

✔ Answers:
14. 4, 16
15. 5, 65
16. 1, 4, 8, 16, 40
17. 1, 2, 6, 8, 24, 36

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Section 4: List all the factors of each number

18. 52
Divisors: 1, 2, 4, 13, 26, 52 → 1, 2, 4, 13, 26, 52

19. 36
1, 2, 3, 4, 6, 9, 12, 18, 36 → 1, 2, 3, 4, 6, 9, 12, 18, 36

20. 57
Check divisibility:
- Odd → not divisible by 2
- Sum: 5+7=12 → divisible by 3 → 57 ÷ 3 = 19
So factors: 1, 3, 19, 57 → 1, 3, 19, 57

21. 63
Sum: 6+3=9 → divisible by 3 → 63 ÷ 3 = 21
63 ÷ 7 = 9 → so 7×9=63
Factors: 1, 3, 7, 9, 21, 63 → 1, 3, 7, 9, 21, 63

22. 96
Even → divisible by 2
Keep dividing:
96 → 48 → 24 → 12 → 6 → 3
So factors:
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
→ 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

✔ Answers:
18. 1, 2, 4, 13, 26, 52
19. 1, 2, 3, 4, 6, 9, 12, 18, 36
20. 1, 3, 19, 57
21. 1, 3, 7, 9, 21, 63
22. 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

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Section 5: Questions 23–24

23. In questions 18–22, which numbers are divisible by:
a) by 2? → Even numbers: 52, 36, 96 → 52, 36, 96
b) by 3? → Sum of digits divisible by 3:
- 52: 5+2=7 → no
- 36: 3+6=9 → yes
- 57: 5+7=12 → yes
- 63: 6+3=9 → yes
- 96: 9+6=15 → yes
→ 36, 57, 63, 96
c) by 2 and 3? → Must be divisible by both → must be even AND sum divisible by 3
→ From above: 36, 96 → 36, 96

✔ Answers:
a) 52, 36, 96
b) 36, 57, 63, 96
c) 36, 96

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Section 6: State the missing factors. Use factors greater than 1

24. $ 2 \times \_ \times \_ = 180 $
→ Divide 180 by 2 → 90
Now factor 90 into two numbers >1 → e.g., 9 × 10 → So: $ 2 × 9 × 10 = 180 $
Other options: 6 × 15, 5 × 18, etc. → Any valid pair
Answer: 9, 10 (or others like 6, 15)

25. $ \_ \times \_ \times \_ = 300 $
Find three factors >1 → Try: 5 × 6 × 10 = 300
Or 5 × 5 × 12 = 300
Or 3 × 5 × 20 = 300
Any valid triplet → Example: 5, 6, 10

26. $ \_ \times 4 \times \_ = 240 $
→ 240 ÷ 4 = 60 → Now factor 60 into two numbers >1
→ 6 × 10 = 60 → So: $ 6 × 4 × 10 = 240 $
Answer: 6, 10

27. $ 8 \times \_ \times \_ = 192 $
→ 192 ÷ 8 = 24 → Factor 24 → 3 × 8 = 24 → So: $ 8 × 3 × 8 = 192 $
But 8 is repeated → OK
Or 4 × 6 → $ 8 × 4 × 6 = 192 $ → Better
Answer: 4, 6

28. $ \_ \times \_ \times 2 = 168 $
→ 168 ÷ 2 = 84 → Factor 84 → e.g., 6 × 14 = 84
So: $ 6 × 14 × 2 = 168 $ → Answer: 6, 14

29. $ \_ \times 9 \times \_ = 225 $
→ 225 ÷ 9 = 25 → 25 = 5 × 5 → So: $ 5 × 9 × 5 = 225 $
Answer: 5, 5

30. $ \_ \times \_ \times \_ = 212 $
Factor 212 → Even → 2 × 106 → 106 = 2 × 53
So: 2 × 2 × 53 = 212 → All >1
Answer: 2, 2, 53

✔ Answers:
24. 9, 10
25. 5, 6, 10 (example)
26. 6, 10
27. 4, 6
28. 6, 14
29. 5, 5
30. 2, 2, 53

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Section 7: Find the smallest number whose factors are...

We need the least common multiple (LCM) of the given numbers.

31. 3, 4, and 5
LCM(3,4,5):
- 3: 3
- 4: 2²
- 5: 5
→ LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60

32. 2, 4, and 9
- 2: 2
- 4: 2²
- 9: 3²
→ LCM = 2² × 3² = 4 × 9 = 36

33. 11, 4, and 2
- 11: 11
- 4: 2²
- 2: 2
→ LCM = 2² × 11 = 4 × 11 = 44

34. 10, 2, and 6
- 10: 2 × 5
- 2: 2
- 6: 2 × 3
→ LCM = 2 × 3 × 5 = 30

✔ Answers:
31. 60
32. 36
33. 44
34. 30

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Section 8: Use divisibility rules

35. 729 → divisible by 8 or 9?

- By 8: Last three digits = 729 → 729 ÷ 8 = 91.125 → Not divisible → No
- By 9: Sum of digits = 7+2+9 = 18 → 18 ÷ 9 = 2 → Yes → Yes

36. 1520 → divisible by 8 or 9?

- By 8: Last three digits = 520 → 520 ÷ 8 = 65 → Yes → Yes
- By 9: Sum = 1+5+2+0 = 8 → not divisible by 9 → No

37. 14112 → divisible by 8 or 9?

- By 8: Last three digits = 112 → 112 ÷ 8 = 14 → Yes → Yes
- By 9: Sum = 1+4+1+1+2 = 9 → 9 ÷ 9 = 1 → Yes → Yes

✔ Answers:
35. Divisible by 9 only
36. Divisible by 8 only
37. Divisible by both 8 and 9

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Section 9: Word Problem

38. There are 24 desks in the classroom. In how many ways can the desks be arranged in equal groups?

We need to find number of factors of 24, because each factor represents a possible group size.

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

So there are 8 ways to arrange them in equal groups.

Example:
- 1 group of 24
- 2 groups of 12
- 3 groups of 8
- 4 groups of 6
- 6 groups of 4
- 8 groups of 3
- 12 groups of 2
- 24 groups of 1

✔ Answer: 8 ways

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✔ Final Summary of Answers:

#### Part A: Missing factors
1. 9
2. 7
3. 7
4. 9
5. 7
6. 8
7. 7
8. 4

#### Part B: Two factor pairs
9. 6×4, 8×3
10. 13×2, 26×1
11. 6×8, 12×4
12. 8×10, 5×16
13. 9×12, 6×18

#### Part C: Complete factor lists
14. 4, 16
15. 5, 65
16. 1, 4, 8, 16, 40
17. 1, 2, 6, 8, 24, 36

#### Part D: All factors
18. 1, 2, 4, 13, 26, 52
19. 1, 2, 3, 4, 6, 9, 12, 18, 36
20. 1, 3, 19, 57
21. 1, 3, 7, 9, 21, 63
22. 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

#### Part E: Divisibility
23a) 52, 36, 96
23b) 36, 57, 63, 96
23c) 36, 96

#### Part F: Missing factors (>1)
24. 9, 10
25. 5, 6, 10
26. 6, 10
27. 4, 6
28. 6, 14
29. 5, 5
30. 2, 2, 53

#### Part G: Smallest number with given factors
31. 60
32. 36
33. 44
34. 30

#### Part H: Divisibility by 8 and 9
35. Divisible by 9 only
36. Divisible by 8 only
37. Divisible by both 8 and 9

#### Part I: Word problem
38. 8 ways

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