What is the last digit of 1/5200?

[fawk3s]

Homework Statement

What would be the last number of 1/5²⁰⁰⁰ as a decimal fraction (or whatever it is for english)?

Homework Equations

The Attempt at a Solution

This is not a homework problem. Its a question in an old maths competition.
I already know the answer is supposed to be 6, but I am interested in how to solve it.

Dont tell me to think about it or demand for any attempted solutions because there arent any.

Thanks in advance,
fawk3s

 

[Mark44]

I don't understand what you are asking. Although you are using a subscript, I think that you might have meant it to be an exponent, so that what you want is the last nonzero digit of 1/(5²⁰⁰⁰). If that's really the problem, start by looking at the decimal representations of 1/5, 1/25, 1/125, 1/625, and so on, and see if you recognize a pattern.

 

[fawk3s]

I don't understand what you are asking. Although you are using a subscript, I think that you might have meant it to be an exponent, so that what you want is the last nonzero digit of 1/(5²⁰⁰⁰). If that's really the problem, start by looking at the decimal representations of 1/5, 1/25, 1/125, 1/625, and so on, and see if you recognize a pattern.

Yes, I made a little mistake in the OP. I edited it now.
But the thing is, I need to find the last number of this number at the state of
0,0000000000000000000534534... (<- these are random numbers inserted).
You are not allowed to use a calculator. How to do it?

 

[Mark44]

Read the rest of my other post. I laid out some things for you to do.

 

[HallsofIvy]

The last digit of 1/5²⁰⁰. One way to do this would be to note that if you set $x= 1/5^{200}$, then $(1/2^{200})x= 1/(10)^{200}= 10^{-200}$. That means that x= $2^{200}(10^{-200})$ and the last digit of x is the "ones" digit of 2²⁰⁰. Now look at powers of 2: 2¹= 2, 2³= 8, 2⁴= 16, 2⁵= 32, 2⁶= 64, 2⁷= 128, 2⁸= 256, 2⁹= 512, 2¹⁰= 1024, 2¹¹= 2048, 2¹²= 4096, etc. The ones digits of those are 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, ... Do you see the pattern? Can you prove that pattern? Now that pattern clearly repeats every 4th time and 200/4= 50 with 0 remainder. If that pattern is correct, 1/5²⁰⁰ has last digit the same as the fourth digit in that sequence, 6.