Pls help me to understand the solution
any positive integer x can be written as (2!)^a*(3!)^b*(5!)^c*(7!)^d
is there any proof for the above line?
Read this : Unique Prime Factorization
Also, the entire statement is:
If it can be written as the form (2!) c 2 * (3!) c 3 * (5!) c 5 * (7!) c 7, there will be only one unique way.
Suppose that there exists two ways to write down x in this form, we can assume that the two ways are (2!) a 2 * (3!) a 3 * (5!) a 5 * (7!) a 7 and (2!) b 2 * (3!) b 3 * (5!) b 5 * (7!) b 7.
We find the largest i such that a i ≠ b i , Then we know there exists at least one prime number whose factor is different in the two ways.
But according to the Fundamental Theorem of Arithmetic, there is only one prime factorization of each integer. So we get a contradiction.
Not all numbers can be represented in that form but if it can be there will be only 1 way.
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