We define is is the exponent of p in the expansion of a.

Thus by definition we directly have the following facts –

1.

2.

3.

Equality when

4.

5.

We also have the legendre’s formula –

6.

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.

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