I cant solve this:
37x + 8 = z
43y + 11 = z
z < 1000
x,y - natural numbers
I know answer is:
x = 21
y = 18
But, i need to prove it.
Can you not prove it by just solving it and showing your work…?
I’m pretty sure that isn’t possible to solve. There are three unknowns and only two sets of data. What is the context? Are you sure there isn’t any more information?
Edit: Sorry, I didn’t realize x and y were natural numbers. Read it but didn’t take it in. @The Lion King has got it.
Set them equal to each other and it becomes the line :
Y = (37/43)x - 3/43
Now the answer has to be an int so the remainder of 37/43 x , must be 3/43
In other words 37/43x mod 1 must be 3/43. and x must also be a positive integer. You can probably stop here but ill continue if needed.
Every time x goes up one, the numerator of 37/43 x mod 1 drops by 3, and it loops around itself (Ex. if x = 1 , the numerator is 4 then if x = 2 the numerator is 41). You can prove this using induction, I wont bother.
every 7 iterations of x makes the numerator of the remainder move up by 1. If x = 1, makes the remainder numerator 37 then x=7 makes the remainder’s numerator 1 according the all the logic I mentioned before.
So lets start at x = 7, the remainders numerator is now 1.
I mentioned before that 7 iterations adds 1 to the numerator, so x = 14 makes the numerator of the remainder 2.
x = 21 makes the numerator 3.
So now plug in 21 to get Y, then verify that both equations equate to a z that is less than 1000.
I hope that was clear 
kind of difficult to explain feel free to ask questions
You have z = 8 (mod 37) and z = 11 (mod 43). Apply the Chinese remainder theorem to find z mod 37*43.