Hello! I can't make a correct proof for H22, can you help me out? Thank you. Also, the answer gave some hints by using theorem 4.4.5 and theorem 7.4.3.

H 22. Define a function g: Z* X Z* - Z* by the formula g(m, n) = 2"3" for all (m, n) E Z XZ. Show that g is one-to-one and use this result to prove that ZT X Z* is countable. CS Scanned with CamScanner22. Hint: Use the unique factorization of integers theorem (Theorem 4.4.5) and Theorem 7.4.3. CS Scanned with CamScannerTheorem 4.4.5 Unique Factorization of Integers Theorem (Fundamental Theorem of Arithmetic) Given any integer n > 1, there exist a positive integer k, distinct prime numbers p1, p2, . . . , pk, and positive integers el', e2, . . . , ek such that n = pi" 19521153 - - -pi*, and any other expression for n as a product of prime numbers is identical to this except, perhaps, for the order in which the factors are written. Scanned With (lamScanner Theorem 7.4.3 Any subset of any countable set is countable. CS Scanned with CamScanner