Show that 10 and 02generate the additive group ZZ7Solutionco

Show that (1,0) and (0,2)generate the additive group Z×Z7.

Solution

consider (x,y) in Z x Z_7

x is in Z

y is in Z_7

since 1 generate Z

so there exists m such that 1.m =1+1+....m times= x

since 2 generates Z_7

so there exists n such that 2.n =2+2+....n times= y

Now

(1,0) + (1,0) + .....m times + (0,2)+ (0,2) + ...ntimes = (x,y)

So (1,0), (0,2) generates every (x,y)