number theory

 Posts: 25
 Joined: Sat Feb 07, 2015 5:40 pm
number theory
Let us define $$S_n$$ to be the set of all integers divisible by $$2014^n$$ but not $$2014^n+^1$$ where $$n$$ is a nonnegative integer. What is the value of $$n$$ (if any) so that $$500!$$ belongs to $$S_n$$.
 seemanta001
 Posts: 13
 Joined: Sat Jun 06, 2015 9:31 am
 Location: Chittagong
Re: number theory
We can observe that $$2014=2\times19\times53$$.Here $53$ is the biggest prime factor of $2014$.
We have to find $\sum\frac{500}{53^n}$ for $n>0$.
There are $9$ numbers that are multiples of $53$ from $1$ to $500$.
No other multiples of $53^n$ are within numbers $1$ to $500$,where $n>1$.
Thus,we get our answer.
We have to find $\sum\frac{500}{53^n}$ for $n>0$.
There are $9$ numbers that are multiples of $53$ from $1$ to $500$.
No other multiples of $53^n$ are within numbers $1$ to $500$,where $n>1$.
Thus,we get our answer.
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