Waste less time on Facebook — follow Brilliant.
×

Number theory

Show that a perfect square cannot end with digits 2,3 or 7?

Note by Goutam Narayan
4 years 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

We look modulo 10.A number can give remainder 0,1,2...9 mod 10.After checking for all of these values, we see that none of them are \(\equiv\) to 2,3 or 7 (mod 10)

Bogdan Simeonov - 4 years ago

Log in to reply

can u plz show it

Goutam Narayan - 4 years ago

Log in to reply

Well ,when a number \( n=10k+r\), then \(n^2=100k^2 + 20kr + r^2 \equiv r^2 \pmod{10}\).Now for

\(r=1, n^2\equiv 1^2 \equiv 1\pmod {10}\)

\(r=2, n^2\equiv 2^2 \equiv 4 \pmod {10}\)

\(r=3, n^2\equiv 3^2 \equiv 9 \pmod{10}\)

\(r=4, n^2\equiv 4^2 \equiv 6 \pmod{10}\)

\(r=5, n^2\equiv 5^2 \equiv 5 \pmod{10}\)

\(r=6, n^2\equiv 6^2 \equiv 6 \pmod{10}\)

\(r=7, n^2\equiv 7^2 \equiv 9 \pmod{10}\)

\(r=8, n^2\equiv 8^2 \equiv 4 \pmod{10}\)

\(r=9, n^2\equiv 9^2 \equiv 1 \pmod{10}\)

Bogdan Simeonov - 4 years ago

Log in to reply

@Bogdan Simeonov thank u very much///

Goutam Narayan - 4 years ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...