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Big Powers

Note by Llewellyn Sterling
2 years, 8 months ago

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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} \( \boxed{123} \)

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We can easily prove that \(3^{444}+4^{333}\) is divisible by \(5\) using Modular Congruences.

Observe that

\[3^{444} \equiv 9^{222} \equiv (-1)^{222} \equiv 1 {\pmod 5}\]

Similarly,

\[4^{333} \equiv (-1)^{333} \equiv -1 {\pmod 5}\]

Adding up congruences, we get

\[3^{444}+4^{333} \equiv 1 + (-1) \equiv 0 {\pmod 5} \quad _\square\]

Kishlaya Jaiswal - 2 years, 8 months ago

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