Theory:

Theorem:
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
 
Explanation:
 
Let us take the composite number \(N\).
 
Decompose the number \(N\) into the product of primes.
 
tree.png
 
Here, the number \(N = x_1 \times x_2\). But, both \(x_1\) and \(x_2\) are again composite numbers. So, factorise it further to obtain a prime number.
 
The prime factors of \(x_1 = p_1 \times p_2\).
 
The prime factors of \(x_2 = p_3 \times p_4\).
 
We get, \(N = p_1 \times p_2 \times p_3 \times p_4\) where \(p_1\), \(p_2\), \(p_3\) and \(p_4\) are all prime numbers.
 
If we have repeated primes in a product, then we can write it as powers.
 
tree1.png
 
In general, given a composite number \(N\), we factorise it uniquely in the form N=p1q1×p2q2×p3q3×...×pnqn where p1,p2,p3,...pn are prime numbers, and q1,q2,q3,...qn are natural numbers.
 
Thus, every composite number can be expressed as a product of primes apart from the order.
Example:
Consider a composite number \(26950\).
 
Let us factor this number using the factor tree method.
 
Factor.gif
 
The prime factor of \(26950\) \(=\) \(2 \times 5 \times 5 \times 7 \times 7 \times 11\).
 
That is, \(26950 = 2 \times 5^2 \times 7^2 \times 11\).
 
Here, a composite number \(26950\) is written as a product of prime numbers.
 
If we change the order of the prime numbers, the answer will also be the same composite number.
 
We can write \(26950 = 2 \times 7^2 \times 5^2 \times 11\) or \(26950 = 11 \times 7^2 \times 5^2 \times 2\).
 
Thus, the prime factorisation of a natural number is unique, except for the order of its factors.
Important!
Recall:
 
HCF \(=\) Product of the smallest power of each common prime factor in the numbers.
 
LCM \(=\) Product of the greatest power of each prime factor involved in the numbers.