Waste less time on Facebook — follow Brilliant.
×

Fast Modular Arithmetic

What is the fastest way to show that \(102^2 \equiv 30\mod {247}\)?

Note by Axas Bit
1 year, 9 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

I suppose one way would be to first note that \(102 = 124 - 22\), and so

\(A = 102^{2} = (124 - 22)^{2} = 124^{2} - 2*22*124 + 22^{2} = 62*248 - 22*248 + 484.\)

Now \(248 \equiv 1 \pmod{247}\) and \(484 = 2*248 - 10 \equiv -10 \pmod{247}\), so

\(A \equiv (62*1 - 22*1 - 10) \pmod{247} \equiv 30 \pmod{247}\).

Brian Charlesworth - 1 year, 9 months ago

Log in to reply

Define "fastest".

In terms of no need to think about what to do: \( 102^2 = 42 \times 247 + 30 \).

Calvin Lin Staff - 1 year, 9 months ago

Log in to reply

'Fastest way' should definitely be checking if 247 divides \(102^2-30\). We find that it is true very fast.

Brilliant Member - 1 year, 9 months ago

Log in to reply

Brian Charlesworth gave a standard answer. Here's an alternative approach:

Notice that \(50^2 = 2500 = 30 + 2470 = 30 + 247(10) \), then \[ 102^2 - 50^2 = (102 - 50)(102 + 50) = 52 \times 152 = 247 \times (\ldots ). \]

This tells us that \(102^2 - 50^2 \equiv 0 \pmod {247} \Rightarrow 102^2 \equiv 50^2 \pmod{247} \Rightarrow 10^2 \equiv 30 \pmod{247} \).

Pi Han Goh - 1 year, 9 months ago

Log in to reply

@Calvin Lin @Pi Han Goh @Otto Bretscher

Axas Bit - 1 year, 9 months ago

Log in to reply

@Brian Charlesworth Any thoughts?

Axas Bit - 1 year, 9 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...