Question

If(a,b)=2 ,(b,4)=2 then (a+b,4)=

Answer 1

Answer:

if (a,4)-2 and(b.4)=2, then (a+b,4)= 4

Suppose the gcd(a+b,4)= d.

Since the gcd(a, 4)=2 and god(b,4)=2, it follows that 2/a and 2/b. Then there exist integers m and n so that, a = 2m and b = 2n.

By substituting a and b into the left hand side of gcd(a+b,4)-d we get gcd(2m+2n,4)= gcd(2(m+n),4)

However m and n ere each odd because if they were even, a and b would also be divisible by 4, then the gcd (a,4) and gcg(b,4) would not equals 2. Thus m+n is even, because the sum of two odd integers is even hence there exits an Integer R such that

m+n = 2r.

Now by substituting in 2r in to gcd(2(m+n),4) We get

gcd(2(2r),4) = gcd(41,4).

So we get gcd (4r,4) =d.

It follows that d=4 because the biggest number lhat divices 4 and a multiple of 4 is 4 so the gcd (a+b,4)=4

I hope it helps you