51a+2b=84c How to find smallest solutions for a,b,c

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51a+2b=84c How to find smallest solutions for a,b,c

Mathematics
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@ParthKohli any ideas?
Diophantine equations. Gotta hate them.
I dont think the question is complete

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Other answers:

Well, do you need positive integer solutions?
Yep. Smallest positive
Oh yeah, how do you define smallest \(a,b,c\)?
a b c are integers. no common factors
did u mean a+b+c to be smallest
that also works
Oh, you need coprime a,b,c.
yep
Can someone explain me what the real question is?
what is the smallest amounts of 51 and 2 required to make a multiple of 84
lol, I have \(a = b = c= 0\). ;-)
perhaps I should have mentioned positive :D
51*2 + 2*33=84*2
Oh wait...I did
@sauravshakya Nailed it.
Note: a must be even
smallest amount of 52 // smallest amount of 2 // small amount of 52 and 2?
\[51a + 2b \equiv 0\pmod{84}\]So yup, \(a\) can't be odd.
That means smallest a / smallest b/ smallest c / smallest a+b ?
8(51)+6(2) = 5(84)
So u want to find minimum value of a+b+c???
yes
But they aren't coprime. :-\
Oh whoops, then I didn't mean coprime As in like a,b,c don't have any common factors across the three of them
It's all good then guys :D

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