The Fundamental Theorem Of Arithmetic states that there is one and only one way of expressing any integer, greater than $1$, as a power of its prime factors.
That is if a number $A$ has prime factors $p_1,\ p_2... p_n$, and if we write:
$A = {p_1}^{e_1} \times {p_2}^{e_2} \times ... {p_n}^{e_n}$, then there is a single fixed set of values of $p_1,\ p_2,\ ... p_n$ and $e_1,\ e_2,\ ... e_n$ which wiil satisfy this condition, and no two distinct integers will have the same representation using its prime factors and their exponents.
Let us take a few examples of this:
The prime factors of $72$ are $2$ which appears $3$ times, and $3$ which appears $2$ times.
Therefore, the only way of writing $72$ in terms of its prime factors is $72 = 2^3 \times 3^2$, there are no other value/s of the prime factors or their exponents that will satisfy this equation, and there is no other integer $N$ that can be represented as $2^3 \times 3^2$.
Similarly, $108$ can be represented using the product of its prime factors in only one way, which is:
where each $p_i$ is a prime number, we can say that $x_i = e_i$ for every $i$
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II. Negative Integers
Fundamental theorem of arithmetic applies only to positive integers, greater than $1$. This theorem does not apply for negative integers.
Let us see why using an example:
We can write the number $-150$ as:
$-150 = (-2)^1 \times 3^1 \times 5^2$ or
$-150 = (2)^1 \times (-3)^1 \times 5^2$
So, there could be multiple representations using negatives numbers. That is why fundamental theorem of arithmetic applies only to positive integers using its positive factors.
$\underline{Example-1:}$
Let us attempt the following problem to understand this concept better:
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