# Chinese remainder theorem (CRT) question, flat shapes

## Homework Statement

By hand, find the 4 square roots of 340 mod 437. (437 = 23 * 19).

## Homework Equations

Chinese remainder theorem (CRT)

## The Attempt at a Solution

So this is the wrong way I did it was first I solved ##x^2 \equiv 340 (\operatorname{mod} 19)## and ##x^2 \equiv 340 \operatorname{mod} 23)## which yielded ##x \equiv \pm 6 (\operatorname{mod} 19)## and ##x \equiv \pm 8 (\operatorname{mod} 23)##.

So I rewrote these as
1) ##x = 6 + 19l##
2) ##x = -6 + 19m##
3) ##x = 8 + 23n##
4) ##x = -8 + 23t##
where ##l,m,n,t## are integers.

And I realized that if you replaced ##x## with ##y## and ##l,m,n,t## with ##z##, then you'd have four 2 dimensional lines and you'd have 4 intersection points... and the problem said 4 solutions so I thought this may be something? (but this lead me to decimal solutions and yeah it didn't get me anywhere...)

But today I realized that ##l,m,n,t## are all different(even though they are just arbitrary integers?) so I can't replace them by 1 variable, basically this is a "5 dimensional system of equation" right?

so a couple questions..
1) We have 4 lines in 5 dimensional space. 1) intersects with 3) at some point and 4) at some point. 2) interacts with 3) at some point and 4) at some point. But our solutions are another 4 other lines that look like##x = 220 + 437j##, where j is an integer, etc. So that seems kind of weird that we're looking for a point of intersection as our solution, and then we turn that point into a line?? What is happening here?

Edit: I guess to answer 1), we find the point of intersection ##p##, and then take ##p \operatorname{mod} 437## to find its congruence class, and that will give us all our solutions.

2) I know two lines intersect at a point (not parallel lines) and two planes intersect at a line (not parallel planes). So do n dimensional flat objects interaction at some n-1 dimensional flat object? How would you define flat?

edit2: I've been looking at the wiki for flat/flatness and there's a lot of words I don't know, maybe a better question would be what tools do I need to define flat?

EDIT3: Ok i think i've answered my own question, flat was not the right word. I meant linear. Given some ##l_1: a_1x_1 + a_2x_2 + ... + a_nx_n = c_1## and ##l_2: b_1x_1 + b_2x_2 + ... + b_nx_n = c_2## where ##a_i , b_i, c_1, c_2## are constants and ##x_i##'s are unknowns. Then our solution is some ##2## dimensional plane. I think my confusion is clear so i'll mark this as solved.. for now...

PS: I did realize that to do this problem, I keep those lines as congruences, and solve them 2 at a time. I was able to work out the 4 solutions: 222, 291, 215, 146 (mod 437).

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