A Surprisingly Interesting Diophantine Equation in Number Theory | Math Olympiad

In this video, I am introducing a nice theorem you can use for most of the diophantine equation and provide step-by-step solution for this diophantine equation using Euclidean algorithm. Diophantine equation is an interesting topic in number theory for you to analyze the equations only with integer coefficients, seeking for integer solutions. Very interesting theorem you can use to solve about 80% of diphantine equations. Come check this out by watching the video until the end, and you will solve diophantine equation easily and faster. More to come! Stay tuned!

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[nasrullahhusnan]

60x+33y=9 is a diophantine equation.
Note that x and y can not have the same sign as RHS is 9. End digit of 60x is 0. If x>0 then y<0 and end digit of 33y should be 1. It means that
y={-7, -17, -27, ...}
60x=9-33y With y=-7 then
60x=9-33(-7)
=9+231
=240 --> x=4
However if x<0 then y>0 and end digit of 33y should be 9. It means that y={3, 13, 23, ...}
33y=9-60x
With y=13 then 429=9-60x --> x=-7
Thus (x, y)={(-7, 13), (4, -7)}

[tsafa]

Wow that's crazy. I didn't expect the answer to be that complicated.

I cheated it.
I wanted to get to 6+3=9
So I backed into (60x.1) + (33x.090909) = 9

Works out good except for tiny rounding error

[domedebali]

I somehow started by dividing everything by 3. thanks for the video

[SrisailamNavuluri]

It is same as 20x +11y=3, dividing throughout by 3.
-2x+22x+11y=3
-2x=3-11k, x=4 when k=1, y=-7
x=4-44k, y=-7+20k is the general solution.

[skwbusaidi]

60x+33y=9
Devide by 3
20x+11y = 3
Using mod 11
9x= 3 ( mod 11)
3x=1 ( mod 11)
3x=12 ( mod 11)
x= 4 ( mod 11)
x = 4+ 11n where n is integer
Sub x in 20x+11y =3
20(4+11n) + 11y =3
80 + 20×11n + 11y = 3
11y = -77+-20×11n
y = -7-20n

[ganeshdas]

Prima facie the equation is satisfied with x= 1/10 & y = 1/11

[crustyoldfart]

Although the given solution is ingenious, it is based on a theorem in number theory which I have not studied enough to have known.
I personally have a love/hate relationship with number theory. This example is hard for me to love, since the way the solution is found appears to be tedious, inelegant and not too far off wild guessing. I'm frankly surprised that it would be considered worthy of inclusion in a math olympiad.