Solving A Nice Diophantine Equation from Romania

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[Tritium]

From the line 3xy=6y+4x, we see that 3 divides 3xy and 6y and so divides 3xy-6y=4x. As 3 is prime and doesn't divide 4, this implies 3 divides x. This reduces the possible values for x-2 from 8 to 3, ignoring the invalid solution x=0.

[SuperYoonHo]

Just the way he teaches is awesome. He sould have been my teacher sooner.

[mcwulf]

Nice question.
I suppose a more thorough answer would look at 3y-4 | 8 and use modulo arithmetic but probably quicker in this case to separately check the 8 possible factors of 8 as you did.

[echandler]

first, exclude (0, 0) from the original equation
(3y-4)(x-2)=8
3y-4 = k, k|8
y=(k+4)/3 = (3+k+1)/3 = 1 + (k+1)/3
k = 2 mod 3 and k|8
k = -1, 2, 8, reject -4 which gives y=0
y = 1, 2, 4
x = -6, 6, 3

[starpawsy]

By inspection rewriting 1/4 as 3/12, and then deeming each added to be 3/24 (==1/8) gives us at least one solution immediately.

[jmiki]

That's why you need rigorous restrictions against which you can check when you get a solution. Or at least you should substitute the values into the original equation. Because now you just dropped the (0, 0) solution without any apparent reason while you said it is a valid solution.

[skerJG]

All these people screaming "(0, 0) isn't a solution!" clearly didn't watch the whole video. It was a valid solution to the equation he had but obviously not to the original equation. He introduced an extraneous solution by multiplying by xy and he clearly knows this because he does not include (0, 0) in the final solution set.

[giuseppemalaguti]

Svolgendo risulta x=2+8/(3y-4), per cui ponendo 3y-4=1, 2, 4, 8 ottengo le uniche soluzioni ammissibili 6, 2 e 3, 4

[mrityunjaykumar]

tou skipped to tell that (0, 0) is not possible as 0 in denominator of original equation..

[boguslawszostak]

it is not a sophisticated solution
x is not equal 0, y is not equal 0
y(3x-6)=4x
y=4x/(3x-6)
for x<0 y<4/3 and x>0 so let us try y=1
3x-6=4x x=-6 so -6, 1 is solution

y=0 is not solution
for x>0
for x=1 y=4/-3 is not integer
for x=2 y*0=8 is no solution
for x=3 y=12/3=4 is solution

for x=4 y=16/6 is not integer
for x=5 y=20/9 is not integer
for x=6 y=24/12=2 is solution

for x>6 y<2 and y>4/3 so is not integer
(-6, 1), (3.4), (6, 2) is the answer

[mconceptz]

Hi! big fan of yours, what app you use as a blackboard?