Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

4x mod 5 = 3x-1 mod 5

Should I do:




according to the theorem, it should be: if a mod 5 = b mod 5

then 5|(a-b), but shouldn't we take the greatest number?

share|cite|improve this question
Hint $\rm\,\ m\mid k\iff m\mid -k\quad $ – Bill Dubuque Dec 9 '12 at 0:03

It makes no difference at all. Suppose that $m\mid a-b$; then there is an integer $k$ such that $a-b=km$. But then $b-a=-(a-b)=-km$, and $-k$ is an integer, so $m\mid b-a$. Thus, the following four statements are all equivalent:

  1. $a\equiv b\pmod m$.
  2. $b\equiv a\pmod m$.
  3. $m\mid a-b$.
  4. $m\mid b-a$.
share|cite|improve this answer

This is good to know:

$$a\equiv b\pmod m \;\iff \;m\mid a-b \;\iff \; m\mid (b-a) \;\iff \; b\equiv a\pmod m \quad\quad\quad\tag{1}$$

$$ \text{and }\;\; m\mid c \quad\iff \quad m\mid -c \tag{2}$$

So what you've written is true, both statements:

$$5\mid (4x-(3x-1))\quad \text{AND}\quad 5\mid((3x-1)-4x)$$ That is: $$5\mid (x+1)\quad \iff \quad 5\mid -(x + 1)$$

So $x \equiv -1 \pmod 5\quad\iff \quad x\equiv 4 \pmod 5$.

Indeed, any $x = 5k + 4$ where $k$ is any integer is a solution: $x \in \{ \cdots, -6, -1, 4, 9, ...\}$.

share|cite|improve this answer
i get 5 = x-1 for the first and 5 = -x-1 for the second – CuriousJoe Dec 8 '12 at 22:58
is there a way to get the right value of x, because i always mix them up and never get the right answer for x? – CuriousJoe Dec 8 '12 at 23:04
im trying to get the smallest positive integer, but i don't always get the right answer for some odd reason – CuriousJoe Dec 8 '12 at 23:10
ah nevermind i can do the opposite if the number is negative – CuriousJoe Dec 8 '12 at 23:46
or i can do +m until i get a desired number – CuriousJoe Dec 8 '12 at 23:47

Your Answer


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.