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Question:
1. Find the common factors of :
(a) 20 and 28 (b) 15 and 25 (c) 35 and 50 (d) 56 and 120
Solution: (a) 20 and 28
Factors of 20: 1, 2, 4, 5, 10 and 20
Factors of 28: 1, 2, 4, 7, 14 and 28
Common factors = 1, 2, 4
(b) 15 and 25
Factors of 15: 1, 3, 5 and 15
Factors of 25: 1, 5 and 25
Common factors = 1, 5
(c) 35 and 50
Factors of 35: 1, 5, 7 and 35
Factors of 50: 1, 2, 5, 10, 25 and 50
Common factors = 1, 5
(d) 56 and 120
Factors of 56: 1, 2, 4, 7, 8, 14, 28 and 56
Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120
Common factors = 1, 2, 4, 8
2. Find the common factors of :
(a) 4, 8 and 12 (b) 5, 15 and 25
Solution: (a) 4, 8 and 12
Factors of 4: 1, 2 and 4
Factors of 8: 1, 2, 4 and 8
Factors of 12: 1, 2, 3, 4, 6 and 12
Common factors = 1, 2, 4
(b) 5, 15 and 25
Factors of 5: 1 and 5
Factors of 15: 1, 3, 5 and 15
Factors of 25: 1. 5 and 25
Common factors = 1, 5
3. Find first three common multiples of :
(a) 6 and 8 (b) 12 and 18
Solution: (a) 6 and 8
Multiple of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72 are multiples of 6.
Multiple of 8: 8, 16, 24, 32, 40, 48, 56, 72 are multiples of 8.
Three common multiples are 24, 48 and 72.
(b) 12 and 18
Multiple of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108 are multiples of 12
Multiple of 18: 18, 36, 54, 72, 90, 108 are multiples of 18
Three common factors are 36, 72 and 108.
4. Write all the numbers less than 100 which are common multiples of 3 and 4.
Solution: Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24 ...
Multiples of 4 are 4, 8, 12, 16, 20, 24...
The first two common multiples of 3 and 4 are 12, 24 which are multiple of 12.
So, all common multiples of 4 and 3 are multiple of 12.
Therefore, common multiples of 3 and 4 less than 100 are 12, 24, 36, 48, 60, 72. 84 and 96.
5. Which of the following numbers are co-prime?
(a) 18 and 35 (b) 15 and 37 (c) 30 and 415
(d) 17 and 68 (e) 216 and 215 (f) 81 and 16
Solution: (a) 18 and 35
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 35: 1, 5, 7, 35
Common factor = 1
Since, their common factor is 1. Hence, the given two numbers are co-prime.
(b) 15 and 37
Factors of 15: 1, 3, 5, 15
Factors of 37: 1, 37
Common factors = 1
Since, their common factor is I. Hence, the given two numbers are co-prime.
(c) 30 and 415
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 415: 1, 5, 83, 415
Common factors = 1, 5
Since, their common factor is not only 1. Hence, the given two numbers are not co-prime.
(d) 17 and 68
Factors of 17: 1, 17
Factors of 68: 1, 2, 4, 17, 34, 68
Common factors = 1, 17
Since, their common factor is not only 1. Hence, the given two numbers are not co-prime.
(e) 216 and 215
Factors of 216: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
Factors of 215: 1, 5, 43, 215
Common factors = 1
Since, their common factor is 1. Hence, the given two numbers are co-prime.
(f) 81 and 16
Factors of 81: 1, 3, 9, 27, 81
Factors of 16: 1, 2, 4, 8, 16
Common factors = 1
Since, their common factor is 1. Hence, the given two numbers are co-prime.
6. A number is divisible by both 5 and 12. By which other number will that number be always divisible?
Solution: The number is divisible by both 5 and 12, so the number will also be divisible by the product of 5 and 12.
5 x 12 = 60.
Therefore, the number always be divisible by 60.
7. A number is divisible by 12. By what other numbers will that number be divisible?
Solution: Factors of 12: 1, 2, 3, 4, 6 and 12.
So, the number also be divisible with 1, 2, 3, 4 and 6.
Answer:
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