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### Congruence Class of 10^n Modulo 11

```Date: 04/14/2003 at 09:16:21
From: Matt
Subject: Congruence class of 10^n modulo 11

What is the congruence class of 10^n modulo 11? Use this to determine
the remainder when 654321 is divided by 11.

What does it mean to say that 10^n mod 11 belongs to a congruence
class?

I know that 654321 =  8 (mod 11) but I don't know how that will help
me in this problem.
```

```
Date: 04/27/2003 at 15:03:00
From: Doctor Nitrogen
Subject: Re: Congruence class of 10^n modulo 11

Hi, Matt:

To find the congruence classes 10^n modulo 11, you can start by
looking at those integers b and r which satisfy, by the Division
Algorithm:

10^n = b*11 + r, where

0 <= r < 11.

For any positive integer m, there are m congruence classes
modulo m. To illustrate: for m = 11 they are those integers
in each of the eleven infinite sets:

congruent to 0 modulo 11: {..., -11, 0, 11, 22, 33, 44, 55, ...}

congruent to 1 modulo 11: {..., 1, 12, 23, 34, 45, 56, ...}

congruent to 2 modulo 11: {..., 2, 13, 24, 35, 46, 57, ...}

congruent to 3 modulo 11: {..., 3, 14, 25, 36, 47, 58, ...}
.
.
congruent to 10 modulo 11: {..., 10, 21, 32, 43, 54, 65, ...}

These are the 11 congruence classes modulo 11.

With a little experimentation, you can find, for each fixed

n = 1, 2, 3, ...

which congruence class each integer

10^n

will belong to. For example,

10^1 can be found in the congruence class 10 modulo 11.

10^2 can be found in the congruence class 1 modulo 11.

10^3 can be found in the congruence class 10 modulo 11.

10^4 can be found in the congruence class 100 modulo 11.

10^5 can be found in the congruence class 1000 modulo 11.

10^6 can be found in the congruence class 10000 modulo 11.

The ten congruence classes modulo 11 form ten equivalence classes
modulo 11.

Now all these above look like

10^n is congruent to 10^s modulo 11,

where s is some integer less than n, and

10^n - 10^s

is divisible by 11.

Is there a continuing pattern here? Try more values for n and then
see if you can conjecture something.

mathematics problem.

- Doctor Nitrogen, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Definitions
College Number Theory

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