We define is is the exponent of p in the expansion of a.
Thus by definition we directly have the following facts –
We also have the legendre’s formula –
This function gets to be seriously useful many more plus it is beautiful !! . Setting such an useful function and observing its properties is super-cool.
So, how can we use it ? – Well , for proving that , it is equivalent to proving that every divisor of is also a divisor of . This is equivalent to proving the fact that the exponent of every prime divisor in is greater than the exponent of the same prime divisor in . This
Ofcourse, by definition , if does not divide then
It gives a way through some very tough otherwise problems like this one in AMM –
Prove that is an integer for any positive integers a,b.