Page 59 - Start Up Mathematics_6

All possible factors of 35 are 1, 5, 7 and 35.

Therefore, common factors of 10, 20 and 35 are 1 and 5.
Example 21: Find the first three common multiples of (a) 4 and 6 (b) 12 and 18

Solution: (a) Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, ...

Multiples of 6 are 6, 12, 18, 24, 30, 36, ...
First three common multiples of 4 and 6 are 12, 24 and 36.

(b) Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, ...
Multiples of 18 are 18, 36, 54, 72, 90, 108, ...

First three common multiples of 12 and 18 are 36, 72 and 108.
Example 22: Write all the numbers less than 100, which are common multiples of 3, 4 and 6.

Solution: Multiples of 3, 4 and 6 which are less than 100 are given below.
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57,
60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99.

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76,
80, 84, 88, 92, 96.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96.

Therefore, common multiples of 3, 4 and 6 (less than 100) are 12, 24, 36, 48, 60,
72, 84 and 96.

Example 23: Which of the following numbers are co-prime?
(a) 9 and 16 (b) 25 and 420 (c) 17 and 68
Solution: (a) All possible factors of 9 are 1, 3 and 9.

All possible factors of 16 are 1, 2, 4, 8 and 16.

Clearly, 1 is the only common factor of 9 and 16. Therefore, 9 and 16 are
co-prime.
(b) All possible factors of 25 are 1, 5 and 25.

All possible factors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 18, 21, 28, 30,
35, 45, 60, 70, 84, 105, 140, 210 and 420.
Clearly, 25 and 420 have common factors other than 1. Therefore, 25 and 420 are
not co-prime.

(c) All possible factors of 17 are 1 and 17.
All possible factors of 68 are 1,4, 17 and 68.

Clearly, 1 and 17 are the common factors of 17 and 68. Therefore, 17 and 68
are not co-prime.
Example 24: A number is divisible by 18. By what other numbers will that number be divisible?

Solution: The number will be divisible by all factors of 18, i.e., 1, 2, 3, 6, 9 and 18. Therefore,
the other numbers are 1, 2, 3, 6 and 9.

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