Let n be a positive integer and let a bZnbe invertible Prove

Let n be a positive integer and let a, bZ_nbe invertible. Prove or disprove each of the following.

(a) ab is invertible

(b) ab is invertible.

Solution

(a) 1 and n 1 are invertible (we know n 1 is invertible since consecutive integers are relatively prime),

but 1 (n 1) = 0 is not invertible. So ab is not invertible.

(b) (a b) (b ¹ a ¹ ) = 1, so a b is invertible (and actually its inverse is b ¹ a ¹ ).