Question

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!

Answer 1

Oops, maybe this involves some trial and error :P

Answer 2

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

Answer 3

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

Answer 4

This works with -8 and -6

Answer 5

If b is -8 then a will be -16

Answer 6

I see a lot of solutions to this :/

Answer 7

Not many, less than 25 I believe ;)

Answer 8

24 or 23 or 22.. I guess

Answer 9

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

Answer 10

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

Answer 11

Oh wait, it goes further.

Answer 12

Your problem is hard.

Answer 13

No spoiler please :)

Answer 14

How do we do it by the way?

Answer 15

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.

Answer 16

is the answer 5? @FoolForMath ?

Answer 17

No Arnab.

Answer 18

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

Answer 19

not 5^^

Answer 20

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

Answer 21

8?

Answer 22

is it right?

Answer 23

No, not right

Answer 24

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

Answer 25

Na na

Answer 26

is all the way wrong or just the 11?

Answer 27

22

Answer 28

I m pretty sur it's 22

Answer 29

24

Answer 30

is 24 correct answer.?

Answer 31

No, you are close by the way

Answer 32

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.

Answer 33

HINT: There are 23 of them.

Answer 34

Real challenge is to find a quick analytic approach :)

Answer 35

yeah i just found the other 12, making 23.

Answer 36

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...

Answer 37

That's right joe! Congrats man!

Answer 38

ty :)