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Chapter 6: linear equations and the greatest common factor (2)

Now, we know the equation ax+by=gcd(a,b) There is always an integer solutionxAndyHow many solutions are there and how to express them?We are mutually exclusiveaAndbLet gogcd(a,b)=1, assuming(x_1,y_1)It’s an equationax+by=1A solution of. adoptx_1subtractbSum of multiples ofy_1addaOther solutions can be obtained. In other words, for any integerkWe get a new solution(x_1+kb,y_1-ka)By calculating a(x_1+kb)+b(y_1-ka)=ax_1+akb+by_1-bka=ax_1+by_1=1 Still observinggcd(a,b)=1It can be proved […]