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  Math problem  (Read 616 times)
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Offline Doubstract
« Posted 2014-03-05 15:27:05 »

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.
Offline opiop65

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« Reply #1 - Posted 2014-03-05 15:28:22 »

Can you not prove it by just solving it and showing your work...?

Offline quew8

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Medals: 33



« Reply #2 - Posted 2014-03-05 22:03:05 »

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.
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Offline The Lion King
« Reply #3 - Posted 2014-03-05 22:54:05 »

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 to want it more than you want to breath, then you will be successful"
Offline pjt33
« Reply #4 - Posted 2014-03-05 23:31:54 »

You have z = 8 (mod 37) and z = 11 (mod 43). Apply the Chinese remainder theorem to find z mod 37*43.
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