# Linear Diophantine Equations: Integer Solutions $x,y$ exist for $ax+by=c$, but how do I find them by hand?

I'm trying to find which of $133x+203y=38$, $133x+203y=40$, $133x+203y=42$, and $133x+203y=44$ have integer solutions. I know that only the third equation suffices for these conditions because $gcd(133,203)=7$ which only divides $42$, but how do I find the solution with smallest possible $x\geq 0$ for $133x+203y=42$?

How do I find solutions $a,b$ to the equation $19a+29b=1$ by hand?

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Since $\gcd(19,29) = 1$, you can use Euclid's algorithm to find $a$ and $b$ such that $19a + 29b = 1$. Then multiply solutions by $6$ to get $x$ and $y$ such that $19x + 29y = 6$ or $133x + 203y = 42$. –  TMM Apr 28 '13 at 23:30
Do you know Euclidean alogorithm? –  ᴊ ᴀ s ᴏ ɴ Apr 28 '13 at 23:30
@TMM Why $6$ Mr./Ms. TMM? –  Trancot Apr 28 '13 at 23:39
@Trancot: If $19a + 29b = 1$ then $19(6a) + 29(6b) = 6(19a + 29b) = 6$ so then $x = 6a$ and $y = 6b$ leads to a solution to $19x + 29y = 6$. –  TMM Apr 29 '13 at 11:39

Correct. Note $\rm\ 19x\!+\!29y=6\, \Rightarrow\, mod\ 19\!:\ y \equiv \dfrac{6}{29}\equiv \dfrac{6}{10}\equiv \dfrac{12}{20}\equiv \dfrac{12}{1}\equiv -7\,\$ so $\rm\,\ \color{#c00}{y = -7\!+\!19n},\:$ therefore $\rm\ x\, =\, \dfrac{6\!-\!29\,\color{#c00}y}{19} = \dfrac{6\!-\!29(\color{#c00}{-7+19n}))}{19}\, =\, 11\!-\!29n.$
Beware $\$ One can employ fractions $\rm\ x\equiv b/a\$ in modular arithmetic (as above) only when the fractions have denominator  coprime  to the modulus  (else the fraction may not (uniquely) exist,  i.e. the equation $\rm\: ax\equiv b\,\ (mod\ m)\:$ might have no solutions, or more than one solution). The reason why such fraction arithmetic works here (and in analogous contexts) will become clearer when one learns about the universal properties of fraction rings (localizations).