Bezout's Identity

ax + by =d

 Integer a Integer b Minimal Bezout Coefficient x Minimal Bezout Coefficient y Write any integer (both positive or negative) k gcd(a,b) d a/d a' b/d b' Minimal Bezout Co-efficient (Maximum Value) < b' |x| Minimal Bezout Co-efficient (Maximum Value) < a' |y| a'x + b'y =1 a'x+b'y Higher Bezout Coefficients: x+(k* b/d) x2 Higher Bezout Coefficients: y-(k* a/d) y2 Working: First fill up a,b and  click press. Then click submit. you shall get value of |x| and |y| which is maximum numerical value of x,y.Try with various values of x,y less than those values till you get a'x+b'y=1. That is the value of one pair of (x,y). Put those values of x,y and put various integer  values of k, you will get various (x2,y2) Example: Let a=12 and b=42; Minimal pair of Bezout co-efficient are(-3,1) & (4,-1). Others are (-10,3),(11,-3),    (18,-5) etc
 Bezout's identity or Bezout's lemma is a theorem that states that if a and b are two nonzero integers and d is the gcd(a,b), then there exist integers x,y such that ax + by =d x,y are called Bezout Coefficients for (a,b). They are not unique. Among various pairs of Bezout Coefficients, exactly 2 pairs satisfy |x| < b/d and |y| < a/d . These are called minimal pairs. This is based on a property of Euclidean division. Once a pair is found, other pairs can be generated by the formula x2= x+(kb/d) and y2= y-(ka/d)