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When there is equation with the same a and b value the solutiob of x and y also will be the same but i got an trouble from this problem.

I have 2 linear diophantine equation :

  1. 2173x + 2491y = 53
  2. 2173x + 2491y = 159

The gcd of 2491 and 2173 are 53.

First Equation is have an solution because 53 is multiple of 53.

Second Equation also have an solution because 159 is multiple of 53.

When i'm start counting with the extended euclidian method to find the x and y value, because both equation have the same a and b value that mean i got the same x and y value, where x = -8 and y = 7 Check : 2173(-8) + 2491(7) = 53

But when it come to the second Equation you Will notice if it will have the same value that is 53 and not 159, why this happened, and how to find the x and y value for the second equation

Bill Dubuque
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Darevil294
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    Welcome to Math SE. Have you tried multiplying both sides of $2173(-8) + 2491(7) = 53$ by $3$? – John Omielan Sep 21 '23 at 03:55
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    I just tried it sir, and the solution -24 and 21 and it right thanks for that sir – Darevil294 Sep 21 '23 at 04:05
  • As in proof $(\Leftarrow)$ here if $,d=\gcd(a,b),$ then a solution of $,ax+by = cd,$ can be found by scaling by $,c,$ any (Bezout) solution of $,a\bar x+b\bar y = d,,$ which yields that $,a(c\bar x) + b(c\bar y) = cd.,$ In modular fraction language, to compute $,x\equiv cd/a\pmod{!b},$ scale $,\bar x\equiv d/a,$ by $:c\ \ $ – Bill Dubuque Sep 21 '23 at 05:06

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