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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 where are prime numbers, and 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.