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  1. A Diophantine equation ax+by = c always has a solution whenever a and b are relatively prime.
  2. Find x ,y such that $$93x-81y=3 $$
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  • $\begingroup$ Do you know Euclidean division algorithm ? $\endgroup$
    – DiffeoR
    Apr 1, 2014 at 13:02
  • $\begingroup$ divide by 3 and you get : 31x - 27y = 1. Now you can solve it ? $\endgroup$
    – DiffeoR
    Apr 1, 2014 at 13:09
  • $\begingroup$ yes i know findind the gcd of two number , then (m,n)=(n,m-tn) for any integer t $\endgroup$
    – noor
    Apr 1, 2014 at 13:11
  • $\begingroup$ @YiyuanLee actually the current statement is valid as well. It doesn't say that if $a$ and $b$ are not relatively prime it will not have solutions. $\endgroup$
    – Guy
    Apr 1, 2014 at 13:12
  • $\begingroup$ how to show the first statment $\endgroup$
    – noor
    Apr 1, 2014 at 13:13

2 Answers 2

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Divide throughout by $3$. You get $31x-27y=1$. Now note that $gcd(31,27)=1$ Thus, by Euclidean algorythm there exist integers $p$ and $q$ such that. $31p+27q=1$. Compare to get values of $x$ and $y$.

You have,

$31=27(1)+4$

$27=4(6)+3$

$4=3(1)+1$

Thus you have

$1=4-3$

$=4-(27-4(6))$

$=4(7)-27$

$=7(31-27)-27$

$=31(7)-27(8)$.

Thus $31(7)+27(-8)=1$

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  • $\begingroup$ I have done this (31, 27) = 1 $\endgroup$
    – noor
    Apr 1, 2014 at 13:14
  • $\begingroup$ Do you know how to write gcd as a linear combination of the two numbers? $\endgroup$
    – Swapnil Tripathi
    Apr 1, 2014 at 13:17
  • $\begingroup$ You have, 31=27*1+4.....27=4*6+3....4=3*1+1....thus you have 1=4-3=4-(27-4*6)=4*7-27=(31-27)*7-27=31*7-27*8. I hope you understand, it's a bit clumsy. $\endgroup$
    – Swapnil Tripathi
    Apr 1, 2014 at 13:22
  • $\begingroup$ write clear solution if posibble $\endgroup$
    – noor
    Apr 1, 2014 at 13:25
  • $\begingroup$ I edited my answer. $\endgroup$
    – Swapnil Tripathi
    Apr 1, 2014 at 13:25
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Infinitely many solutions of the same.

$y=31k+8$

$x=27k+7$

or

$y=31k-23$

$x=27k-20$

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  • $\begingroup$ what is the value of k it satisfies the initial solution $\endgroup$
    – noor
    Apr 4, 2014 at 18:16
  • $\begingroup$ y = -8-93t and x = 7+81t now cjose the value of t which satisfies the initial equation $\endgroup$
    – noor
    Apr 4, 2014 at 18:17
  • $\begingroup$ what is the value of t which satisfies the 93x+81y=3 $\endgroup$
    – noor
    Apr 4, 2014 at 19:07
  • $\begingroup$ Your question, I do not quite understand. Number "k" - can be any course. What we want. $\endgroup$
    – individ
    Apr 5, 2014 at 4:14

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