Page 51 - Start Up Mathematics_6

Composite numbers
Numbers having more than two factors are called composite numbers. These numbers have at
least one factor other than 1 and the number itself. Since 4 has more than two factors, i.e., 1, 2
and 4, therefore it is a composite number. Similarly, 6, 8, 9, 10 ... are also composite numbers.

1 is neither prime nor composite.

Twin primes

If the difference between any two prime numbers is 2, then such pairs of prime numbers are called
twin primes. Examples of twin primes are (3, 5), (5, 7), (11, 13), etc.

Sieve of Eratosthenes

In third century BC, a Greek mathematician named Eratosthenes gave a simple method to find
prime numbers from 1 to 100. A stepwise procedure is given below.

Prime Numbers from 1 to 100 by Sieve Method

Step 1: List all the natural numbers from 1 to 100 in a tabulated form as shown.
Step 2: Cross out 1 as it is not a prime number.

Step 3: Encircle the first prime number 2 and cross out all its multiples except 2 itself,
i.e., cross out 4, 6, 8, ... up to 100.

Step 4: Encircle the next uncrossed number 3 and cross out all its multiples except 3
itself. Ignore the previously crossed out numbers like 6, 12, 18, ... .
Step 5: The next uncrossed element is 5. Encircle it and cross out all the multiples of 5
except 5 itself. Ignore the previously crossed out numbers.

Step 6: Continue this process till you find that all the numbers from 1 to 100 are either
encircled or crossed out. All encircled numbers are prime numbers and all
crossed out numbers, other than 1 are composite numbers.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

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