"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
on modulo \(16\) graph paper have some points in common. What is the sum of the -coordinates of those points?
By the Euclidean Algorithm, the y-intercept is (7,18), so the answer is 7 + 18 = 25.
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?
hm... what do you mean by edit my answer further. I don't quite understand
Parts that I believe were cut off?:
"where\(0\le x" Thank you for confirming :) "What is the sum of the -coordinates of those points?" ( So I need to know if we want to know the sum of the x or y coordinates. ) Alright, I think you're good to go.
by the way, the part where it says \(0\le x it is acually \(0\le x < m\)
So the answeris just 1 + 9?