Page 58 - Start Up Mathematics_6

9. Using divisibility tests, determine which of the following numbers are divisible by 11:

(a) 4,55,400 (b) 10,00,001 (c) 91,30,825
10. Write the smallest digit in the blank space of each of the following numbers so that the
number formed is divisible by 3: (a) 3,2__,386 (b) 5__,945
11. Write a digit in the blank space of each of the following numbers so that the number formed
is divisible by 9: (a) 24,1__,279 (b) 5,94,__84
12. Write a digit in the blank space of each of the following numbers so that the number formed
is divisible by 11: (a) 9,32,__38 (b) 8,__9,487

Common Factors
A number which exactly divides two or more given numbers is the common factor of the given
numbers. For example, let’s find common factors of 15 and 18.

The factors of 15 are 1 , 3 , 5 and 15.
The factors of 18 are 1 , 2, 3 , 6, 9 and 18.

The common factors of 15 and 18 are 1 and 3.

Common Multiples

A number which is exactly divisible by two or more given numbers is the common multiple of
the given numbers. For example, let’s find the common multiples of 15 and 18.
Multiples of 15 are 15, 30, 45, 60, 75, 90 , 105, 120, 135, 150, 165, 180 , ...

Multiples of 18 are 18, 36, 54, 72, 90 , 108, 126, 144, 162, 180 , ...
Therefore, common multiples of 15 and 18 are 90, 180, ....

Co-prime Numbers
Two numbers having only 1 as the common factor are called co-prime numbers. 7 and 11,
3 and 5, 4 and 15, ..., etc. are examples of co-prime numbers as 1 is the only common factor
between them.

• A pair of prime numbers is always co-prime.
• A pair of co-prime numbers need not necessarily contain prime numbers.
Example 20: Find the common factors of:

(a) 30 and 64 (b) 56 and 120 (c) 10, 20 and 35
Solution: (a) All possible factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.

All possible factors of 64 are 1, 2, 4, 8, 16, 32 and 64.
Therefore, common factors of 30 and 64 are 1 and 2.
(b) All possible factors of 56 are 1, 2, 4, 7, 8, 14, 28 and 56.

All possible factors of 120 are 1, 2, 3, 4, 6, 8, 10, 12, 15, 20, 30, 40, 60 and 120.
Therefore, common factors of 56 and 120 are 1, 2, 4 and 8.

(c) All possible factors of 10 are 1, 2, 5 and 10.
All possible factors of 20 are 1, 2, 4, 5, 10 and 20.

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