Hi!

I have such task:

Demonstrate, that, (a;b)=(a;a+b)

There is no more explanation. Teacher said that there is somthing to do with number sharing properties.

Please help!

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- Nov 4th 2012, 05:06 AM #1

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## Re: (a;b)=(a;a+b)

And it didn't occur to you to

**tell**us that to begin with? sigh.

If n= (a; b), the greatest common divisor of a and b, since n**is**a common divisor, then a= xn and b= yn. a+ b= xn+ yn= (x+y)n so that n**is**a common divisor of a and a+ b. Now, suppose there were a larger common divisor- that is there exist m such that a= mj, a+ b= mk for integers j and k and m> n. The b= (a+ b)- a= mk- mj= m(k- j) so that m is also a common divisor of b.