(a;b)=(a;a+b)

Jan 2012
25
0
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!
 
Jan 2008
484
109
UK
What have you been studying in class?

On its own your assignment is meaningless.
 
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Plato

MHF Helper
Aug 2006
22,507
8,664
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!
With so little given there is no way to help you. Sorry.
 
Jan 2012
25
0
a and b are natural numbers, I suppose.
And those brackets mean the greatest common divisor of the numbers inside.
 
Last edited:

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
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.