# Modular arithmetic

• Feb 28th 2010, 09:03 AM
integral
Modular arithmetic
$\displaystyle \textrm{I\; am\; trying\; to\; learn\; modular\; arithmetic,\; and\; this\; right \; here\; stumped\; me.}$
$\displaystyle 10\cdot(-13)+27\cdot 6\equiv-122\mathrm{mod}2$
$\displaystyle \textrm{I\; thought\; It\; would\; go\; like\; this:}$
$\displaystyle 10\cdot(-13)+27\cdot 6\; \; \; \textrm{Can be reduced to in mod}2$
$\displaystyle 0\cdot(1)+1\cdot 0=0\mathrm{mod}2$

$\displaystyle \textrm{Please explain this to me someone}$
• Feb 28th 2010, 10:45 AM
Bacterius
Hi Integral,

Suppose you have :
$\displaystyle 10 \times (-13) + 27 \times 6$

This can be written as :

$\displaystyle (2 \times 5 + 0) \times (2 \times (-6) + 1) + (2 \times 13 + 1) \times (2 \times 3 + 0)$

Note that modulo two, adding two doesn't change the residue, and multiplying by two makes the residue zero. So this simplifies to :

$\displaystyle 0 \times 1 + 1 \times 0$ (mod 2)

And therefore the whole expression is equivalent to zero modulo 2 (it is even). These are just modular arithmetic theorems that need to be learnt (that adding the modulus to terms doesn't change anything, multiplying by the modulus makes the term zero, that the residue of a sum of terms is the same as the sum of the residues of each term, ... then you got the powers and Fermat's/Euler's Theorem ...)

:)
The text passage contains a mathematical question related to modular arithmetic, and an answer is provided by another user. The question is about understanding a specific modular arithmetic calculation, and the answer explains the simplification process and the application of modular arithmetic theorems.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Modular arithmetic

• Feb 28th 2010, 09:03 AM
integral
Modular arithmetic
$\displaystyle \textrm{I\; am\; trying\; to\; learn\; modular\; arithmetic,\; and\; this\; right \; here\; stumped\; me.}$
$\displaystyle 10\cdot(-13)+27\cdot 6\equiv-122\mathrm{mod}2$
$\displaystyle \textrm{I\; thought\; It\; would\; go\; like\; this:}$
$\displaystyle 10\cdot(-13)+27\cdot 6\; \; \; \textrm{Can be reduced to in mod}2$
$\displaystyle 0\cdot(1)+1\cdot 0=0\mathrm{mod}2$

$\displaystyle \textrm{Please explain this to me someone}$
• Feb 28th 2010, 10:45 AM
Bacterius
Hi Integral,

Suppose you have :
$\displaystyle 10 \times (-13) + 27 \times 6$

This can be written as :

$\displaystyle (2 \times 5 + 0) \times (2 \times (-6) + 1) + (2 \times 13 + 1) \times (2 \times 3 + 0)$

Note that modulo two, adding two doesn't change the residue, and multiplying by two makes the residue zero. So this simplifies to :

$\displaystyle 0 \times 1 + 1 \times 0$ (mod 2)

And therefore the whole expression is equivalent to zero modulo 2 (it is even). These are just modular arithmetic theorems that need to be learnt (that adding the modulus to terms doesn't change anything, multiplying by the modulus makes the term zero, that the residue of a sum of terms is the same as the sum of the residues of each term, ... then you got the powers and Fermat's/Euler's Theorem ...)

:)
user
The text passage contains a mathematical question related to modular arithmetic, and an answer is provided by another user. The question is about understanding a specific modular arithmetic calculation, and the answer explains the simplification process and the application of modular arithmetic theorems.

TYPE: SINGLE_QUESTION_WITH_ANSWER