"Modulo m graph paper" consists of a grid of m^2 points, representing all pairs of integer residues (x,y) where $0\le x

Question

"Modulo m graph paper" consists of a grid of m^2 points, representing all pairs of integer residues (x,y) where $0\le x<m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\equiv x^2\pmod 5$ would consist of the points $(0,0)$, $(1,1)$, $(2,4)$, $(3,4)$, and $(4,1)$.

The graphs of $$y\equiv 5x+2\pmod{16}$$ and $$y\equiv 11x+12\pmod{16}$$ on modulo $16$ graph paper have some points in common. What is the sum of the $x$-coordinates of those points?

rational · Answer

use ` $ some latex mess$ ` to write inline latex

h0pe · Answer

sorry I use that format for other sites let me fix it

h0pe · Answer

"Modulo m graph paper" consists of a grid of m^2 points, representing all pairs of integer residues (x,y) where $$0\le x<m$$. To graph a congruence on modulo m graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $$y\equiv x^2\pmod 5$$ would consist of the points (0,0), (1,1), (2,4), (3,4), and (4,1).

The graphs of $$y\equiv 5x+2\pmod{16}$$ and $$y\equiv 11x+12\pmod{16}$$ on modulo 16 graph paper have some points in common. What is the sum of the x-coordinates of those points?

rational · Answer

so you're given a congruence system : 
$$y\equiv 5x+2\pmod{16}\~\y\equiv 11x+12\pmod{16}$$

and asked to find the sum of x values of the solutions

rational · Answer

simply subtract both the congruences so that "y" disappears

h0pe · Answer

basically just 11x+12-5x+2?

h0pe · Answer

or 6x+10 (mod 16)

rational · Answer

Yes all that equals 0 in mod 16

rational · Answer

because left hand side is 0

rational · Answer

$$y\equiv 5x+2\pmod{16}\~\y\equiv 11x+12\pmod{16}$$


subtracting you get
$$0\equiv 6x+10\pmod{16}$$

rational · Answer

which is same as 
$$6x\equiv -10\pmod{16}$$

rational · Answer

same as
$$6x\equiv 6\pmod{16}$$

h0pe · Answer

so do I now find the inverse of 6?

h0pe · Answer

mod 16

rational · Answer

there wont be an inverse for 6 in mod 16

h0pe · Answer

then what..?

rational · Answer

maybe try finding it, you will see why there is no inverse

h0pe · Answer

ok

h0pe · Answer

then how do I solve it?

rational · Answer

simply divide 6 through out recalling the congruence property : 

$$\large  ax\equiv a \pmod{n} ~\implies ~  x \equiv 1 \pmod{\frac{n}{\gcd(n,a)}}$$

rational · Answer

$$6x\equiv 6\pmod{16}~ \implies~ x\equiv 1\pmod{8}$$

rational · Answer

so solutions to given system are $x=8k+1$ where  $k\in \mathbb{Z}$

rational · Answer

since you want only the solutions between $0\le x\lt 16$, you get two x values :
$$x=1, ~9$$

rational · Answer

add them up

h0pe · Answer

10, thank you!

rational · Answer

yw