Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose I have $a\equiv b \text{ (mod $c$)}$ and I just know $c$. I want to know, say $b$. Is this the same as $a$ mod $c$? If so, why?

I think I confuse the congruence with the equality symbol, because only recently I learned that sometimes $mod$ is used as an operator, and not as a mere part of the congruence symbol.

share|improve this question
add comment

2 Answers

up vote 3 down vote accepted

One of the possible values for $b$ would be $a$ mod $c$, but there will be others too, because adding any multiple of $c$ to a solution yields another solution.

For example: $6 \equiv b$ (mod 3). We have (6 mod 3) = 0, and one of the possible values for $b$ is 0, but actually $b$ could be any multiple of 3.

Most of the time, in number theory, you don't really want to think about mod as an operator; you want to think of it as defining a relationship between numbers (i.e., a congruence). The congruence symbol is deliberately made to look like an equality symbol because many of the properties of equality also apply to congruences. For example, you can add, subtract, or multiply any integer to both sides of a congruence, and the congruence remains valid. (Just be careful with division.)

These properties are most cleanly stated when stated in terms of congruences, rather than using mod as an operator.

share|improve this answer
add comment

Actually $a\equiv b\text{ ( mod $c$)}$ means $a-b=ck$, where $k\in\mathbb{Z}$. That is when a and b are divided by c in both cases we have same remainder.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.