"Modulo graph paper" consists of a grid of \(m^2\) points, representing all pairs of integer residues \((x,y)\) where\(0\le x . 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 \((0,0),(1,1),(2,4),(3,4),(4,1)\) .

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 -coordinates of those points?

Imcool Nov 27, 2022

#1**0 **

By the Euclidean Algorithm, the y-intercept is (7,18), so the answer is 7 + 18 = 25.

Guest Nov 27, 2022

#4**0 **

5x + 2 = 11x + 12 (mod 16)

6x = -10 (mod 16)

6x = 6 (mod 16)

3x = 3 (mod 8)

Inverse of 3 in mod 8 is 3.

x = 1 (mod 8)

believe that the range for x are [0, m) based on my inferences. If I am wrong, let me know.

Then, all x values that satisfy this equation in those boundaries are 1 and 9.

Hence, the coordinates of the points are (1, 7), (9, 15).

Now, can you solve the question from here?

Voldemort Nov 27, 2022

#5