+0  
 
-1
70
10
avatar+124 

"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?

 Nov 27, 2022
 #1
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0

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

 Nov 27, 2022
 #2
avatar+124 
-1

That makes sense, thank you!

 Nov 27, 2022
 #3
avatar+124 
-1

can you show me the solution by any chance? Not just the answer!

 Nov 27, 2022
 #4
avatar+285 
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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?

 Nov 27, 2022
edited by Voldemort  Nov 27, 2022
edited by Voldemort  Nov 27, 2022
 #5
avatar+124 
-1

hm... what do you mean by edit my answer further. I don't quite understand

 Nov 27, 2022
 #7
avatar+285 
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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. 

Voldemort  Nov 27, 2022
edited by Voldemort  Nov 27, 2022
edited by Voldemort  Nov 27, 2022
 #6
avatar+124 
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by the way, the part where it says \(0\le x  it is acually \(0\le x < m\)

 Nov 27, 2022
 #8
avatar+124 
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So the answeris just 1 + 9?

 Nov 27, 2022
 #10
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0

Yes, the answer is 1 + 9 = 10.

 

--V

Guest Nov 27, 2022
 #9
avatar+124 
-1

its the x cooradinates the answer wants

 Nov 27, 2022

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