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Playing With Numbers

    Prime and Composite Numbers

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6th class > Playing With Numbers > Prime and Composite Numbers

Prime and Composite Numbers

We are now familiar with the factors of a number.

Observe the number of factors of a few numbers arranged in this table.

NumbersFactorsNumber of Factors
111
21, 22
31, 32
41, 2, 43
51, 52
61, 2, 3, 64
71, 72
81, 2, 4, 84
91, 3, 93
101, 2, 5, 104
111, 112
121, 2, 3, 4, 6, 126

We find that (a) The number 1 has only one factor (i.e. itself ). (b) There are numbers, having _exactly two factors 1 and the number itself. Such number are 2, 3, 5, 7, 11 etc. These numbers are prime numbers.

Try to find some more prime numbers other than these.

(c) There are numbers having more than two factors like 4, 6, 8, 9, 10 and so on.

These numbers are composite numbers

Is 15 a composite number? Why? What about 18? 25?

is neither a prime nor a composite number.

Without actually checking the factors of a number, we can find prime numbers from 1 to 100 with an easier method.

The Sieve of Eratosthenes

It turned out to be quite difficult to determine if a number is prime: you always had to find all its prime factors, which gets more and more challenging as the numbers get bigger. Instead, the Greek mathematician Eratosthenes of Cyrene came up with a simple algorithm to find all prime numbers up to 100: the Sieve of Eratosthenes.

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First we need to write down all numbers up to 100.
We know that 1 is not prime, so we delete it.
The smallest prime number is 2. Any multiple of 2 can’t be prime, since it has 2 as a factor. Therefore we can cross out all multiples of 2.
The next number in our list is 3 – again a prime number. All multiples of 3 can’t be prime, since they have 3 as a factor. Therefore we can cross these out as well.
The next number, 4, is already crossed out so we move on to 5: it is a prime number and again we cross out all multiples of 5.
The next prime number must be , since 6 is crossed out. Once more, we cross out all of its multiples.
The next prime number is . Notice, however, that all of its multiples are . The same is actually true for all other remaining numbers. Therefore all these remaining numbers must be prime.

Now we can count that, in total, there are prime numbers less than 100.

1. Write all the prime numbers less than 15.

By observing the Sieve Method, we can easily write the required prime numbers as , , , , and .

Observe that 2 × 3 + 1 = 7 is a prime number. Here, 1 has been added to a multiple of 2 to get a prime number. Can you find some more numbers of this type?

2 × 5 + 1 = that is

2 × 8 + 1 = that is

2 × 9 + 1 = that is

2 × 15 + 1 = that is