PRIME AND COMPOSITE NUMBERS

Think about the situation :

The teacher gave 5 buttons to David and 6 buttons to Lily and asked them to arrange the buttons in all possible rows in such a way that the number of buttons in each row is equal. They did it, in different ways as shown below :

David's Ways

By arranging 5 buttons in a row, he gets 1 row. 

1 x 5  =  5

By arranging 1 button in each row, he gets 5 rows. 

5 x 1  =  5

He realizes that 5 buttons can be arranged in only 2 rectangular ways. Hence, the only factors of 5 are 1 and 5 (number of rows).

Lily's Ways

By arranging 6 buttons in a row, she gets 1 row. 

1 x 6  =  6

By arranging 3 buttons in each row, she gets 2 rows.

2 x 3  =  6

By arranging 2 buttons in each row, she gets 3 rows. 

3 x 2  =  6

By arranging 1 button in each row, she gets 6 rows. 

6 x 1  =  6

She realizes that 6 buttons can be arranged in 4 rectangular ways. Hence, the factors of 6 are 1, 2, 3 and 6 (number of rows).

●Make different rectangular arrangements using 1 button, 2 buttons, 3 buttons, 4 buttons,…, 10 buttons and complete the following table :

From the table, we conclude that

● A natural number greater than 1, having only two factors namely 1 and the number itself, is called a prime number.

For example, 2 (1 x 2) is a prime number as is 13 (1 x 13).

● A natural number having more than 2 factors is called composite number. 

For example, 15 is a composite number (15 = 1 x 3 x 5) as is 70 (1 x 2 x 5 x 7).

Perfect Number

A number is a perfect number if the sum of its factors other than the given number gives the same number.

For example, 6 is a perfect number, since adding the factors of 6 (other than 6), namely 1, 2 and 3 gives the given  number 6.

That is, 1 + 2 + 3  =  6 is the given number.

Check whether 28, 54 and 496 are perfect numbers or not.

Co-Primes or Relatively Prime

If the greatest common factor (GCF) of two numbers is 1, then the numbers are said to be co-primes or relatively prime or mutually prime.

Here, the two numbers can both be primes as (3, 11) or both can be composites as (16, 35) or one can be a prime and other a composite as (7, 14).

Activity

(i) List out the prime and composite numbers represented by the dates in the month of October.

(ii) Generate a few composite numbers by product of two or more natural numbers.

(iii) Classify the numbers 34, 57, 71, 93, 101, 111 and 291 as prime or composite.

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