Page 39 - Maths Class 06

Co-prime Numbers
Two numbers having only 1 as the common factor are called co-prime numbers. 7 and 11, 3 and 5,
4, 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.

EX AM PLE 5. Which of the fol low ing num bers are co-prime?
(a) 9 and 16 (b) 25 and 420 (c) 17 and 68
SOLUTION : (a) 9 and 16
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) 25 and 420

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, 20, 21, 28, 30, 35, 42,
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) 17 and 68
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.

The Sieve of Eratosthenes
Prime numbers and their properties were first studied by the
ancient Greek mathematicians. The Greek Eratosthenes 1 2 3 4 5 6 7 8 9 10
devised a method to find the prime numbers commonly 11 12 13 14 15 16 17 18 19 20
known as ‘The Sieve of Eratosthenes’. This sieve drains out
21 22 23 24 25 26 27 28 29 30
all the composite numbers and leaves behind the prime
numbers. 31 32 33 34 35 36 37 38 39 40

To use the Sieve of Eratosthenes, make a chart of first 100 41 42 43 44 45 46 47 48 49 50
natural numbers. Cross out 1 as it is not prime. Circle the 51 52 53 54 55 56 57 58 59 60
smallest even prime number, i e. . 2. Cross all the multiples of
61 62 63 64 65 66 67 68 69 70
2. Circle 3, the next prime number and cross all the
multiples of 3, then circle the next prime 5 and again cross 71 72 73 74 75 76 77 78 79 80
all the multiples of 5. Continue this process till all the 81 82 83 84 85 86 87 88 89 90
numbers are either crossed out or circled.
91 92 93 94 95 96 97 98 99 100
It can be clearly seen that all crossed numbers (except 1) are
composite numbers and circled numbers are prime numbers.

Do you observe any pattern in the numbers, (3, 5), (5, 7), (11, 13), (17, 19), (29, 31) and so on.
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