Math proof question! not fun at all :(

[Hoeboy]

Prove g.c.d(a,b)=g.c.d(-a,-b).

g.c.d stands for greatest common divisor. So it's basically asking to prove that the g.c.d of two positive number should be the same as the g.c.d of those same two number, but negative, not positive.

 

[Vadatajs]

I may have misunderstood the question but:


the only difference between a and -a is a factor of -1. As long as there is another factor of both a and b that's greater than -1 (ie, all factors for 13, positive and negative are {-13, -1, 1, 13} (-1 * -13 = 13), -13 has the exact same factors. 26 would be {-26, -13, -2, -1, 1, 2, 13, 26}, as would -26, so if a=13 and b=26, the gcd of a,b and -a,-b would both be 13, because 13 > -1 > -13.

 

[aux]

Originally posted by: Hoeboy
Prove g.c.d(a,b)=g.c.d(-a,-b).

g.c.d stands for greatest common divisor. So it's basically asking to prove that the g.c.d of two positive number should be the same as the g.c.d of those same two number, but negative, not positive.

This question is kind of strange because usually people define gcd only for non-negative integers (mainly due to historical reasons), and then use the equality that you want to prove (as well as another one, similar to this and a symmetry argument) as definition for gcd's eventually involving negative integers.

You may have defined gcd using something like the Euclid's algorithm. If that's the case, the proof should be just comparing the respective lines of Euclid's algorithm for computing the two gcs.