## FoolForMath 3 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. FoolForMath
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. FoolForMath

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. FoolForMath

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

11. ParthKohli

Oh wait, it goes further.

12. ParthKohli

Your problem is hard.

13. FoolForMath

No spoiler please :)

14. ParthKohli

How do we do it by the way?

15. FoolForMath

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. Arnab09

is the answer 5? @FoolForMath ?

17. ParthKohli

No Arnab.

18. Arnab09

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. Arnab09

not 5^^

20. Arnab09

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. rfig.khalil

8?

22. rfig.khalil

is it right?

23. ParthKohli

No, not right

24. rfig.khalil

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

25. ParthKohli

Na na

26. rfig.khalil

is all the way wrong or just the 11?

27. rfig.khalil

22

28. rfig.khalil

I m pretty sur it's 22

29. rfig.khalil

24

30. rfig.khalil

is 24 correct answer.?

31. ParthKohli

No, you are close by the way

32. joemath314159

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. FoolForMath

HINT: There are 23 of them.

34. FoolForMath

Real challenge is to find a quick analytic approach :)

35. joemath314159

yeah i just found the other 12, making 23.

36. joemath314159

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. FoolForMath

That's right joe! Congrats man!

38. joemath314159

ty :)