# Modular Arithmetic

• Dec 12th 2010, 01:55 PM
extatic
Modular Arithmetic
Hi Guys,

Having some trouble with basic modular arithmetic, most of the materials i have dont seem to be helping..

a simple question 2 mod 10 = 2.. how did they get = 2?

3 mod 18 = 3, could you please explain the process
• Dec 12th 2010, 02:08 PM
tonio
Quote:

Originally Posted by extatic
Hi Guys,

Having some trouble with basic modular arithmetic, most of the materials i have dont seem to be helping..

a simple question 2 mod 10 = 2.. how did they get = 2?

3 mod 18 = 3, could you please explain the process

Definition $a=b\!\!\pmod n\Longleftrightarrow n\mid (a-b)\Longrightarrow a-b=kn$ , when all the letters

here symbolize integer numbers.

Thus, $2=2\!\!\pmod{10}$ because $10\mid(2-2=0)\,,\,\,0=10\cdot 0$ , and etc.

A less trivial example: $11=47\!\!\pmod{18}\,\,because\,\,11-47=-36=18\cdot (-2)$

Tonio
• Dec 12th 2010, 02:15 PM
extatic
Quote:

Originally Posted by tonio
Definition $a=b\!\!\pmod n\Longleftrightarrow n\mid (a-b)\Longrightarrow a-b=kn$ , when all the letters

here symbolize integer numbers.

Thus, $2=2\!\!\pmod{10}$ because $10\mid(2-2=0)\,,\,\,0=10\cdot 0$ , and etc.

A less trivial example: $11=47\!\!\pmod{18}\,\,because\,\,11-47=-36=18\cdot (-2)$

Tonio

Thank you Tonio,

what if b (mod n) is only given, not a ??
• Dec 12th 2010, 02:45 PM
tonio
Quote:

Originally Posted by extatic
Thank you Tonio,

what if b (mod n) is only given, not a ??

$b\!\!\pmod n$ hasn't much meaning beyond indicating that we're considering an element b modulo some integer n...

Tonio
• Dec 12th 2010, 03:00 PM
Petek
I'd like to add to tonio's reply. Programmers use b (mod n) to represent the remainder upon dividing b by n. For example, 7 (mod 5) equals 2, since 7/5 = 1, with remainder 2. See , for example, section 3.4 of Concrete Mathematics by Graham, Knuth and Patashnik.
• Dec 12th 2010, 03:10 PM
extatic
Thank you to you both :)

Petek thank you for that..
• Dec 12th 2010, 09:53 PM
extatic
If this helps anyone,

2648 (mod 7)

2648 / 7 = 378.28...........
378 x 7 = 2646

2648 - 2646 = 2

2468 (mod 7) = 2

thanks for the help guys