### 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.

Here, the number $$N = x_1 \times x_2$$. But, both $$x_1$$ and $$x_2$$ are again a composite number. 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.

In general, given a composite number $$N$$, we factorise it uniquely in the form $N={{p}_{1}}^{{q}_{1}}×{{p}_{2}}^{{q}_{2}}×{{p}_{3}}^{{q}_{3}}×...×{{p}_{n}}^{{q}_{n}}$ where ${p}_{1},\phantom{\rule{0.147em}{0ex}}{p}_{2},\phantom{\rule{0.147em}{0ex}}{p}_{3},\phantom{\rule{0.147em}{0ex}}...\phantom{\rule{0.147em}{0ex}}{p}_{n}$ are prime numbers, and ${q}_{1},\phantom{\rule{0.147em}{0ex}}{q}_{2},\phantom{\rule{0.147em}{0ex}}{q}_{3},\phantom{\rule{0.147em}{0ex}}...\phantom{\rule{0.147em}{0ex}}{q}_{n}$ 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.

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, then also the answer will 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.
Significance of the fundamental theorem of arithmetic
1. If a prime number $$p$$ divides $$ab$$, then $$p$$ divides either $$a$$ or $$b$$. That is, $$p$$ divides at least one of them.
Example:
Let us take a prime number $$3$$ divides $$5 \times 6$$.

$\frac{5×6}{3}$

Here, $$3$$ cannot divide $$5$$, but it divides $$6$$.

That is, a prime number $$p$$ divides at least one of them.

2. If a composite number $$n$$ divides $$ab$$, then $$n$$ neither divide $$a$$ nor $$b$$.
Example:
Let us take a  composite number $$4$$ divides $$2 \times 6$$.

$\frac{2×6}{4}$

Here, $$4$$ neither divide $$2$$ nor divide $$6$$. But, it divides the product of $$2 \times 6 = 12$$.

Thus, if a composite number $$n$$ divides $$ab$$, then $$n$$ neither divide $$a$$ nor $$b$$.
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.