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7. A number is divisible by 12. By what other numbers will that number be divisible?

Since the number is divisible by 12, it will also be divisible by its factors i.e., 1, 2, 3, 4, 6, 12. Clearly, 1, 2, 3, 4, and 6 are numbers other than 12 by which this number is also divisible.

6. A number is divisible by both 5 and 12. By which other number will that number be always divisible?

Factors of 5=1,5  Factors of 12=1,2,3,4,6,12 As the common factor of these numbers is 1, the given two numbers are coprime and the number will also be divisible by their product, i.e. 60, and the factors of 60=1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

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

(a) 18 and 35  Factors of 18=1,2,3,6,9,18  Factors of 35=1,5,7,35  Common factor =1  Therefore, 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  Therefore, 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  As these numbers have a common factor...

4. Write all the numbers less than 100 which are common multiples of 3 and 4.

Multiples of 3 = 3,6,9,12,15…  Multiples of 4 = 4,8,12,16,20…  Common multiples = 12,24,36,48,60,72,84,96

3. Find first three common multiples of :
(a) 6 and 8        (b) 12 and 18

(a) 6 and 8  Multiple of 6=6,12,18,24,30…  Multiple of 8=8,16,24,32…… common multiples =24,48,72    (b) 12 and 18  Multiples of 12=12,24,36,78  Multiples of 18=18,36,54,72 3 common multiples = 36,72,108

2. Find the common factors of :
(a) 4, 8 and 12      (b) 5, 15 and 25

(a) 4,8,12  Factors of 4=1,2,4  Factors of 8=1,2,4,8  Factors of 12=1,2,3,4,6,12  Common factors =1,2,4    (b) 5,15, and 25  Factors of 5=1,5  Factors of 15=1,3,5,15  Factors of 25=1,5,25  Common factors =1,5

1. Find the common factors of :
(a) 20 and 28      (b) 15 and 25      (c) 35 and 50      (d) 56 and 120

(a) 20 and 28 Factors of 20=1,2,4,5,10,20  Factors of 28=1,2,4,7,14,28  Common factors =1,2,4     (b) 15 and 25 Factors of 15=1,3,5,15  Factors of 25=1,5,25  Common factors =1,5   (c) 35 and 50    Factors of 35=1,5,7,35  Factors of 50=1,2,5,10,25,50  Common factors =1,5   (d) 56 and 120 Factors of 56=1,2,4,7,8,14,28,56  Factors of 120=1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120  Common factors =1,2,4,8
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