The only factor that is common to 200 and an integer n is 1

The only factor that is common to 200 and an integer n is 1. When n/200 is written as a decimal number, what is the minimum number, of digits to the right of the decimal point?

Solution

In order to check the number of digits after decimal point on dividing n/200 where n is an integrer and GCD(n,200)=1 we can check the number of digits after decimal point on dividing n/200 where 1<=n<200 and GCD(200,n) = 1

We know that there are 80 such integers n (between 1 and 200) for which GCD(200, n) = 1. Therefore, we check the number of digits after decimal point on dividing n by 200 for these 80 numbers. The one for which the number of digits after decimal point are least, will be our answer.

Furthermore, for any integer n>200 such that GCD(200,n) = 1 we will have the digits after decimal point repeating as for the previous 80 cases. Therefore, we do not need to check of the number greater than 200 (also for negative numbers).

Coming to the 80 numbers between 1 and 200 such that GCD(200,n) = 1, we get the number of digits after decimal point equal to 3. Therefore, there are three digits after decimal point on dividing any relative prime number by 200. Hence the answer for this question is 3.