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Theorem:

**Every composite number can be expressed (factorised) as a product of primes, and this factorization 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 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 a power.

In general, given a composite number \(N\), we factorise it uniquely in the form $N={{p}_{1}}^{{q}_{1}}\times {{p}_{2}}^{{q}_{2}}\times {{p}_{3}}^{{q}_{3}}\times ...\times {{p}_{n}}^{{q}_{n}}$ 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, 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 factorization 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.