Fundamental Theorem of Arithmetic
We have discussed about Euclid Division Algorithm in the previous post.
Fundamental Theorem of Arithmetic:
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.
Example: (i) 6615 = 3 × 3 × 3 × 5 × 7 × 7
= 33 × 5 × 72
(ii) 532400 = 2 × 2 × 2 × 2 × 5 × 5 × 11 × 11 × 11
Fundamental Theorem of Arithmetic Example Problems With Solutions
Example 1: 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.
Sol. Since, 6 = 2 × 3; 6n = 2n × 3n
⇒ The prime factorisation of given number 6n
⇒ 6n is not divisible by 7.
Example 2: 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.
Sol. 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.