FoolForMath
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!
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ParthKohli
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Oops, maybe this involves some trial and error :P
ParthKohli
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What is \(\mathbb{Z}\) in here?
ParthKohli
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This works with -8 and -6
ParthKohli
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If b is -8 then a will be -16
ParthKohli
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I see a lot of solutions to this :/
FoolForMath
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Not many, less than 25 I believe ;)
ParthKohli
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24 or 23 or 22.. I guess
ParthKohli
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The minimum of b is I guess -36
maximum is 84
FoolForMath
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Don't guess, try to form a analytic approach :)
ParthKohli
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Oh wait, it goes further.
ParthKohli
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Your problem is hard.
FoolForMath
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No spoiler please :)
ParthKohli
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How do we do it by the way?
FoolForMath
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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.
Arnab09
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is the answer 5? @FoolForMath ?
ParthKohli
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No Arnab.
Arnab09
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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
Arnab09
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not 5^^
Arnab09
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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
rfig.khalil
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8?
rfig.khalil
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is it right?
ParthKohli
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No, not right
rfig.khalil
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the equation is equivalent to (x-8)*(y+12)=-96 so the answer is 11
ParthKohli
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Na na
rfig.khalil
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is all the way wrong or just the 11?
rfig.khalil
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22
rfig.khalil
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I m pretty sur it's 22
rfig.khalil
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24
rfig.khalil
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is 24 correct answer.?
ParthKohli
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No, you are close by the way
joemath314159
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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.
FoolForMath
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HINT: There are 23 of them.
FoolForMath
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Real challenge is to find a quick analytic approach :)
joemath314159
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yeah i just found the other 12, making 23.
joemath314159
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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...
FoolForMath
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That's right joe! Congrats man!
joemath314159
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ty :)