## anonymous 4 years ago Fool's problem of the day, How many ordered pair of $$(x,y)$$ are there such that $$x, y \in \mathbb{Z}$$ and $$\frac 2 x - \frac 3 y = \frac 1 4$$ Good luck!

1. ParthKohli

Oops, maybe this involves some trial and error :P

2. ParthKohli

What is $$\mathbb{Z}$$ in here?

3. anonymous
4. ParthKohli

This works with -8 and -6

5. ParthKohli

If b is -8 then a will be -16

6. ParthKohli

I see a lot of solutions to this :/

7. anonymous

Not many, less than 25 I believe ;)

8. ParthKohli

24 or 23 or 22.. I guess

9. ParthKohli

The minimum of b is I guess -36 maximum is 84

10. anonymous

Don't guess, try to form a analytic approach :)

11. ParthKohli

Oh wait, it goes further.

12. ParthKohli

13. anonymous

14. ParthKohli

How do we do it by the way?

15. anonymous

It's problem of the day, it's supposed to be interesting. However my approach takes less than 10 seconds ;) I will post it later though.

16. anonymous

is the answer 5? @FoolForMath ?

17. ParthKohli

No Arnab.

18. anonymous

okay, we can get after simplification.. x=8y/(y+12) as x is an integer, 8y must be a multiple of y+12 8y=2*2*2*y so, here are possible equations y+12=1 y+12=2 y+12=4 y+12=8 y+12=2y y+12=4y y+12=8y as y +12 is a factor of 8y all the equations except the last one will satisfy for an integral value of y so, the answer is.. 6 ordered pairs

19. anonymous

not 5^^

20. anonymous

sorry, there will be other options too if RHS is negative y+12=-1 y+12=-2 ................. and so on.. for y to be an integer, there are another 5 solutions so, total 11

21. anonymous

8?

22. anonymous

is it right?

23. ParthKohli

No, not right

24. anonymous

the equation is equivalent to (x-8)*(y+12)=-96 so the answer is 11

25. ParthKohli

Na na

26. anonymous

is all the way wrong or just the 11?

27. anonymous

22

28. anonymous

I m pretty sur it's 22

29. anonymous

24

30. anonymous

31. ParthKohli

No, you are close by the way

32. anonymous

I was able to find 11 ordered pairs that are solutions: (7, 84) (6,36) (5,20) (4,12) (2,4) (-4,-4) (-8,-6) (-16,-8) (-24,-9) (-40,-10) (-88,-11) I think there might be more, but im missing an idea.

33. anonymous

HINT: There are 23 of them.

34. anonymous

Real challenge is to find a quick analytic approach :)

35. anonymous

yeah i just found the other 12, making 23.

36. anonymous

Most people probably got that:$8y-12x-xy = 0$There is a way you can factor this to get an idea of how many solutions there are...

37. anonymous

That's right joe! Congrats man!

38. anonymous

ty :)