Solving equations with mod

Question

So, I'm trying to solve the following equation using regular algebra, and I don't think I'm doing it right: $3x+5 = 1\pmod {11}$

I know the result is $x = 6$, but when I do regular algebra like the following, I do not get 6:

$3x=1 - 5\pmod{11}$

$x = \dfrac{(1 \pmod{11} - 5)} 3$

So, I figured that since $1 \pmod{11} = 1$ the equation becomes

$x = \dfrac{-4} 3$

Which is not 6! I am totally lost here and would appreciate any help......

Answer 1

$3x+5 = 1(mod 11)$ means that $3x+5-1=11n$ for some $n\in \Bbb Z$, and hence $x=\frac{11n-4}{3}$. So $n=6 (x=6) ,n=11(x=17) \cdots$

Notice that $n$ must satisfy that 3 divides $11n-4$. So the solution is not unique.

Answer 2

The division in $\mathbb{F_{11}}$ is not like the division in $\mathbb{Q}$ !!! If you want to find $3(mod \; 11)^{-1}$ think of which element of $\mathbb{Z_{11}}$ multiplied by $3$ gives $1 \; (mod \; 11)$? It's $4$. So $$3x \equiv -4 \; (mod \; 11) \Leftrightarrow x \equiv -4*4 \; (mod \; 11)$$ And $-16 \equiv 6 \; (mod \; 11)$, because $11$ divides $-16 - 6 = -22$.

Answer 3

When we are working with congruence, $\text{mod } n$, for some $n$ (here, 11), we are interested in integral relations, i.e., relationships between integer values, and instead of division, we use the fact, in this case, that $3x +5= 11k + 1$, where $k$ is any integer (the quotient), and $1$ is the remainder when $3x +5$ is divided by $11$

$$3x + 5 \equiv 1\pmod {11} \implies 3x+5 -1\equiv 1-1 = 0 \pmod{11}\iff 3x+4 \equiv 0 \pmod{11}$$

So we are looking for $x$ such that $3x+4 = 11k+ 0 = 11k$ for some integer $k$. That means we need for $\dfrac{3x+4}{11}$ to equal the integer $k$. Note that the given solution is one such solution: When $x=6$, then $3x+4 = 18+4 = 22 = 2(11)$. So when $x = 6$, $11$ divides $3x+4$ evenly, with no remainder.

There are infinitely many solutions $x$ that will meet this requirement, (test out $x = 6-11, x = 6+ 11= 17, x=6+ 2(11)$, etc.) So sometimes people use a representative congruence class $\overline x$, where $x$ is the least non-negative integer solution, and $$\overline x = \{6\pm 11k\mid k\in \mathbb Z\}$$

Answer 4

The congruence relation 3x+5≡1 (mod 11) is equivalent to 3x-11y= -4 where x,y ∈Z.

My solution to the problem may be found in the link to the PDF document: http://www.aespen.ca/AEnswers/1415545790.pdf.

The following image was created from the PDF document above.