Nine teens

The last three digits of \(19^{n}\)is \(001\). Define the \[\color{BLUE}{\huge{second}}\] smallest solution for \(n\), or else to prove that it is impossible.

Note by Bryan Lee Shi Yang
3 years, 3 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

If \(19^n \) ends in \( 001 \), then \( 19^n - 1 \) ends in \( 000 \),i.e, \( 19^n - 1 \) is divisible by \( 1000 \). Or simply, \( 19^n \equiv 1 \pmod{1000} \). By Euler's theorem, \( 19^{\phi(1000)} \equiv 1 \pmod{1000} \) gives us an upper bound of \( \phi(1000) = 400 \).

We can do better though. We see that \( 19^2 \equiv 3^2 \equiv 1 \pmod{8} \). Also, \( 19^{\phi(125)} \equiv 19^{100} \equiv 1 \pmod{125} \). So we get a smaller solution \( 100 \).

At this point, I can't find a better solution. Wolfram Alpha gives the smallest solution as \( n = 50k \) where \( k \) is an integer, so the answer to your question is \( 100 \).

Siddhartha Srivastava - 3 years, 3 months ago

Log in to reply

19= 20 - 1 . when we do any even power of (20-1) it will end up with last digit 1 succeeded by 3 zeroes if n is a multiple of 50 (binomial expansion) non multiple may give 1 or 2 zeroes before 1 but multiple of 50 gives 3 zeroes . so ans is 100.

Sanyam Sood - 3 years, 1 month ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...