## FoolForMath Group Title 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! 2 years ago 2 years ago

1. ParthKohli Group Title

Oops, maybe this involves some trial and error :P

2. ParthKohli Group Title

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

3. FoolForMath Group Title
4. ParthKohli Group Title

This works with -8 and -6

5. ParthKohli Group Title

If b is -8 then a will be -16

6. ParthKohli Group Title

I see a lot of solutions to this :/

7. FoolForMath Group Title

Not many, less than 25 I believe ;)

8. ParthKohli Group Title

24 or 23 or 22.. I guess

9. ParthKohli Group Title

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

10. FoolForMath Group Title

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

11. ParthKohli Group Title

Oh wait, it goes further.

12. ParthKohli Group Title

13. FoolForMath Group Title

14. ParthKohli Group Title

How do we do it by the way?

15. FoolForMath Group Title

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 Group Title

is the answer 5? @FoolForMath ?

17. ParthKohli Group Title

No Arnab.

18. Arnab09 Group Title

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 Group Title

not 5^^

20. Arnab09 Group Title

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 Group Title

8?

22. rfig.khalil Group Title

is it right?

23. ParthKohli Group Title

No, not right

24. rfig.khalil Group Title

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

25. ParthKohli Group Title

Na na

26. rfig.khalil Group Title

is all the way wrong or just the 11?

27. rfig.khalil Group Title

22

28. rfig.khalil Group Title

I m pretty sur it's 22

29. rfig.khalil Group Title

24

30. rfig.khalil Group Title

31. ParthKohli Group Title

No, you are close by the way

32. joemath314159 Group Title

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 Group Title

HINT: There are 23 of them.

34. FoolForMath Group Title

Real challenge is to find a quick analytic approach :)

35. joemath314159 Group Title

yeah i just found the other 12, making 23.

36. joemath314159 Group Title

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 Group Title

That's right joe! Congrats man!

38. joemath314159 Group Title

ty :)