Question

Let us define the superfactorial !n! of a natural n number as
follows:
!n! := 1!2! · · · n!
What is the maximum power of 2 that divide !$2^{5}$!?

Answer 1

Answer:

The maximum power of 2 is 447 .

Step-by-step explanation:

The superfactorial !n! of a number is defined as !n! = 1! * 2!  ........ * n!

We need to find the maximum power of 2 that does divide !2^5!

!2^5!

> !32!

> 1! * 2! * 3! * ...... * 32!

We will now use the Legendre Formula to individually get the maximum powers of 2, for each of the terms and add them .  

Let us generalize the Legendre formula :

Suppose that for a certain prime number p and for a positive integer k , f(k) represents the exponent of the highest power of p dividing k ;

Then ,

$f(k!) = summation_{1 \: to \: infinity } floor [ k/p^i]$

Finding individually and adding, we get p =2 and k = 447 .

Thus, the required answer is 447.

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