# modular arithmetic, solving ax + b = c (mod d)?

For example 151x - 294 is congruent to 44 (mod 7). How would I go about solving that?

The answer says to simplify it into the ax = b (mod c) form first, but I have no clue on how to get rid of the 294... help?

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possible duplicate of How to find solutions of linear Diophantine ax + by = c ? – Arturo Magidin Apr 2 '11 at 4:54
Note that solving $ax\equiv b\pmod{d}$ is equivalent to solving $ax+dy = b$ with $x$ and $y$ integers. – Arturo Magidin Apr 2 '11 at 4:54
The first thing you should do is simplify all the coefficients modulo $7$; $151\equiv 4\pmod{7}$, $294\equiv 0 \pmod{7}$, and $44\equiv 2\pmod{7}$. So your problem is exactly the same as $4x\equiv 2\pmod{7}$. But if you had not been lucky enough to get that constant term to mod out to $0$, to go from $ax-c\equiv b\pmod{d}$, just add $c$ to both sides of the congruence! – Arturo Magidin Apr 2 '11 at 5:01
@meiryo: There's certain rules (you cannot cancel factors unless they are relatively prime to the modulus; you can only add/multiple congruences modulo the same modulus), but yes. Didn't you see the basic properties before you tried to solve a congruence? – Arturo Magidin Apr 2 '11 at 5:03
@meiryo: I would say they are pretty darn lousy if you have to solve a congruence and you don't know that you can add and subtract the same thing from both sides of the congruence. – Arturo Magidin Apr 2 '11 at 5:11

"a congruent to b mod m" is also written $a \equiv b \pmod m$ and I define it to mean that $a + (m) = b + (m)$ where $(m) = \{ k m \in \mathbb Z | k \in \mathbb Z \}$ is the set of all multiples of $m$ also called the ideal generated by $m$.

What this means, for example, is that $9 \equiv 23 \pmod 7$ since $9 + (7) = \{\cdots,-12,-5,2,9,16,23,30,\cdots\} = 23 + (7).$ In general if number $x = mk+r$ then $x \equiv r \pmod m.$

A general result about relations is that if $R$ is an equivalence relation on $X$ and $f : Y \mapsto X$ is a function then the relation defined by $aSb :\iff f(a) R f(b)$ is also an equivalence relation (on $Y$). This proves that equivalence mod m is an equivalence relation since the relation $R$ is $=$ (which we know to be an equivalence) and the function $f$ is just $+ (m)$.

Since we have an equivalence relation we can do equational reasoning with it. For example:

If $$151x - 294 \equiv 44 \pmod 7$$ then since $$44 \equiv 2 \pmod 7$$ we have $$151x - 294 \equiv 2 \pmod 7,$$ furthermore $$151 \equiv 4 \pmod 7$$ and $$294 \equiv 0 \pmod 7$$ so we have $$4x \equiv 2 \pmod 7.$$ Cancelling out $2$ from both sides gives $$2 x \equiv 1 \pmod 7.$$ To solve this we can just try $x = 1$, $x = 2$, ... until we find a solution.

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Thanks! This goes to prove from what Arturo taught me I've solved this correctly :D – meiryo Apr 2 '11 at 6:22

to solve $151x - 294 \equiv 44 \pmod{7}$, first add 294 on both sides to obtain $151x \equiv 338 \pmod{7}$. now before solving for x, let's first check if an integer solution exists. $(151,7) = 1$, and $1|338$, so x is solvable. so now we are solving for x in $151x + 7m = 338$

now use extended eculid algorithm to solve for x. precisely, set up 2 columns and and try to get the column under "e" to become 338 by row addition, subtraction and multiplication. the third column "m" is not necessary, but I listed it anyway for clarity.

e | x(coeff. of 151) | m(coeff. of 7)

151 | 1 | 0

7 | 0 | 1

147 | 0 (2nd row* 21, and we ignore the m column from now on)

4 | 1 (row1 - row3)

3 | -1 (row2-row4)

1 | 2 (from previous two rows)

338 | 676

so x = 676, but x can be any element in $[676]_{7}$

hope this helps, and sorry about the format. \tabular doesn't seem to work here and I don't know what else to do...

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I got x = 4 using something similar to quanta's method, what's [676]sub7? – meiryo Apr 2 '11 at 6:20
4 works too, since $[676]_{7} = [4]_{7}$. $[676]_{7}$ is a congruence class. in this case, it's the set of all integers of the form $676 + 7m$ for any integer $m$. now you can see $[676]_{7} = [4]_{7}$ because 676 + 7(-96) = 4. so x can be $4+7m$ for any integer m. – user8969 Apr 2 '11 at 6:40