Problem 1
Prove that for every irrational real number a, there are irrational real numbers b and b' such that a+b and ab' are rational, while ab and a+b' are irrational.
Step 3 of 5: Prepare two candidates for b prime
Detailed analysis
Both candidates are irrational, since a nonzero rational divided by an irrational number is irrational. Their products with a are 1 and 2, hence rational.