If a b1 prove that ab ab 1 or 2 Hint Suppose that dab ab Sho

If (a, b)=1, prove that (a+b, a-b) =1 or 2.

Hint: Suppose that d=(a+b, a-b). Show that d|2b, d|2a, and use the result of: If c > 0, prove that (ac, bc)=c(a,b).

Now we will go by using the hint:

Now lets prove that (ac,bc) = c (a,b).

Let (ac ,bc) = m

so we have ac=mx and bc =my ,since m is the G.C.D , we can say that x and y are co primes ( Numbers whose G.C.D is 1).

Now let (a,b) =k,

then a=gk

and b = fk where f,g are co primes.

multiplying with c , we have,

ac=cgk =g *ck

and bc= cfk .= f* ck

Since g and f are co primes , the G.C.D of ac, bc will be ck

So (ac,bc) =ck =c (a,b).

Hence we can say (2a,2b) =2(a,b) = 2 *1 =2.

Now lets prove this:

Let d = (a+b, a-b)

so let a+b =dk -- Equation 1 and

a-b =dp -- Equation 2 , where k and p are co primes

adding both equations we get;

2a= dk+dp =d(k+p) =dx (say)

so 2a = dx ,hence we can say that d divides 2a i.e d|2a.

Now let us subtract the two equations , we have;

2b =dk-dp =d (k-p) =dy (say)

so 2b = dy ,hence we can say that d divides 2b i.e d|2b.

So we have 2a=dx and 2b = dy

Now since k and p are co primes ,

so (2a,2b) = (dx,dy) =d(x,y) = 2 (as we got above)

so d * (x,y) = 2*1

Hence there are two cases , 1. d=2 and (x,y) =1

2. d=1 and (x,y) =2.

Hence d can be either 1 or 2.