The Euclidean Algorithm and Diophantine Equations
... equation ax + by c has no solution. Proof: Let d gcd(a,b). Then there are integers r and s such that dr a and ds b. By way of contradiction, assume that ax + by c does have a solution xo, yo. Then c axo + byo drxo + dsyo. But this says that d|c since c d(rxo + syo). Since this is a c ...
... equation ax + by c has no solution. Proof: Let d gcd(a,b). Then there are integers r and s such that dr a and ds b. By way of contradiction, assume that ax + by c does have a solution xo, yo. Then c axo + byo drxo + dsyo. But this says that d|c since c d(rxo + syo). Since this is a c ...
numbers : rational, irrational or transcendental
... The number system of mathematics begins with the counting numbers 1, 2, 3, . . . which are called natural numbers and is denoted by N. If m, n ∈ N then we cannot always solve the equation x + m = n in N and so from N we arrive at the set of integers Z in which we can solve the above type of equation ...
... The number system of mathematics begins with the counting numbers 1, 2, 3, . . . which are called natural numbers and is denoted by N. If m, n ∈ N then we cannot always solve the equation x + m = n in N and so from N we arrive at the set of integers Z in which we can solve the above type of equation ...
