Let us take any collection of prime numbers, say 2,3,7,11 and 23. If we multiply some are all of these numbers allowing them to repeat as many times as we wish,we can produce infinetly many large positive integers. Let us observe a few:

2 × 3 × 11 = 66

3 × 7 × 11 × 23 = 5313

2² × 3 × 7³ = 8232

Now,Let us suppose your collection of prime includes all the possibles primes.What is your guess about the size of this collections? Does it contain only a finitity number of primes of infinitely many? In fact, there are infinetely many primes.So if we multiply all possibles ways we will get an infinity collections of composite numbers.

Now, let us consider the converse of this statement i.e. if we take a composite number can it be written as a product of prime numbers? The following Therom answers the questions.

Theorem-1.2:

This gives Funedamental Theorem of Arthamatic which says that every composite number can be factorised as a prodect of a prime numbers in a "unique" way except for the order in which the primes occures. For examples when we factorised 210, we regard 2×3×5×7 as same as 3×5×7×2, or any other possible order in which these primes are written. That is given any composite number there is one and only one way to write it as a product of primes as long as we are not particular about the other in which the primes occur.

In generally given a composite number x we factroize it as x=p₁. p₂. p₃.....p_n,where p₁, p₂, p₃....., p_n are primes and written in acensding order,i.e.,p₁≤ p₂≤.......≤p_n. If we combine that equal primes we will get powers of primes. Once we have diceded that the order will be ascending,then the way the number is ffactorized, is unique.For example,

27300=2×2×3×5×5×7×13=2²×3×5²×7×13

Do this

Express 2310 as a product prime factors. Also see how your friends have factorized the number. Have they done it same as you? Verify your final product with your friend's result.Try this for 3 or 4 numbers.What do you conclude.