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!

- anonymous

- katieb

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- ParthKohli

Oops, maybe this involves some trial and error :P

- ParthKohli

What is \(\mathbb{Z}\) in here?

- anonymous

http://en.wikipedia.org/wiki/Integer

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## More answers

- ParthKohli

This works with -8 and -6

- ParthKohli

If b is -8 then a will be -16

- ParthKohli

I see a lot of solutions to this :/

- anonymous

Not many, less than 25 I believe ;)

- ParthKohli

24 or 23 or 22.. I guess

- ParthKohli

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

- anonymous

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

- ParthKohli

Oh wait, it goes further.

- ParthKohli

Your problem is hard.

- anonymous

No spoiler please :)

- ParthKohli

How do we do it by the way?

- 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.

- anonymous

is the answer 5? @FoolForMath ?

- ParthKohli

No Arnab.

- 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

- anonymous

not 5^^

- 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

- anonymous

8?

- anonymous

is it right?

- ParthKohli

No, not right

- anonymous

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

- ParthKohli

Na na

- anonymous

is all the way wrong or just the 11?

- anonymous

22

- anonymous

I m pretty sur it's 22

- anonymous

24

- anonymous

is 24 correct answer.?

- ParthKohli

No, you are close by the way

- 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.

- anonymous

HINT: There are 23 of them.

- anonymous

Real challenge is to find a quick analytic approach :)

- anonymous

yeah i just found the other 12, making 23.

- 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...

- anonymous

That's right joe! Congrats man!

- anonymous

ty :)

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