# Mathematics

### Chapter : Real Numbers

#### Fundamental Theorem of Arithmatic

Statement : Every composite number can be decomposed as a product prime numbers in a unique way, except for the order in which the prime numbers occur.

For example :
(i) 30 = 2 × 3 × 5, 30 = 3 × 2 × 5, 30 = 2 × 5 × 3 and so on.
(ii) 432 = 2 × 2 × 2 × 2 × 3 × 3 × 3 = 24 × 33 or 432 = 33 × 24.
(iii) 12600 = 2 × 2 × 2 × 3 × 3 × 5 × 5 × 7 = 23 × 32 × 52 × 7

In general, a composite number is expressed as the product of its prime factors written in ascending order of their values. Some examples are

(i) 6615 = 3 × 3 × 3 × 5 × 7 × 7 = 33 × 5 × 72
(ii) 532400 = 2 × 2 × 2 × 2 × 5 × 5 × 11 × 11 × 11

Example : Consider the number 6n, where n is a natural number. Check whether there is any value of n ∈ N for which 6n is divisible by 7.

Solution: Since, 6 = 2 × 3; 6n = 2n × 3n
↠ The prime factorisation of given number 6n
6n is not divisible by 7. (Ans)
Ex.16 Consider the number 12n, where n is a natural number. Check whether there is any value of n ∈ N for which 12n ends with the digit zero.
Solution: We know, if any number ends with the digit zero it is always divisible by 5.
↠ If 12n ends with the digit zero, it must be divisible by 5.
This is possible only if prime factorisation of 12n contains the prime number 5.
Now, 12 = 2 × 2 × 3 = 22 × 3
↠ 12n = (22 × 3)n = 22n × 3n
i.e., prime factorisation of 12n does not contain the prime number 5.
↠ There is no value of n ∈ N for which 12n ends with the digit zero. (Ans)