Help needed in NT!

My doubt has been resolved by Satyajit Mohanty and Pi Han Goh.\[\]How would you go about solving this problem?:\[10^n \equiv 2\pmod{19};n \in \mathbb{Z}^{+}\\ Smallest\ such\ n=?\]Please help!

Note by Adarsh Kumar
2 years, 7 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

@Satyajit Mohanty @Pi Han Goh

Adarsh Kumar - 2 years, 7 months ago

Log in to reply

The smallest \(n\) is \(17\). Study residue systems modulo \(n\).

Satyajit Mohanty - 2 years, 7 months ago

Log in to reply

You need to check that 17 is the smallest, though.

Otto Bretscher - 2 years, 7 months ago

Log in to reply

Kindly tell the method!

Adarsh Kumar - 2 years, 7 months ago

Log in to reply

@Adarsh Kumar Oh,wait I got it:did you do it this way:\[10^{18}\equiv1\pmod{19}\\ from\ Euler's\ totient\ function\\ 10^{17}\equiv\dfrac{1}{10}\equiv\dfrac{2}{20}\equiv2\pmod{19}\]?

Adarsh Kumar - 2 years, 7 months ago

Log in to reply

@Adarsh Kumar You have only shown that n = 17 is a solution but you didn't show that it's the smallest solution.

Pi Han Goh - 2 years, 7 months ago

Log in to reply

@Pi Han Goh How do we do that?Please help!

Adarsh Kumar - 2 years, 7 months ago

Log in to reply

@Adarsh Kumar Show that the residues of a prime is (prime - 1). So if you found anything less than a prime, then it's minimum.

Pi Han Goh - 2 years, 7 months ago

Log in to reply

@Adarsh Kumar Yup.

Satyajit Mohanty - 2 years, 7 months ago

Log in to reply

@Satyajit Mohanty Ok,thanx!Are you in mood for a trigo problem?

Adarsh Kumar - 2 years, 7 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...