12. Use the method of exhaustion to prove the following statements:For every prime numberp <18, p-9 is not divisible by 10.pcan be one of the following numbers: 2, 3, 5, 7, 11, 13, and 17. So,p-9 can be oneof these numbers: -7, -6, -4, -2, 2, 4, and 8. Since none of these integers are divisibleby 10, the proof is complete.13. Prove the following statements by the method of indirect proof (proof by contradiction):(a)∀r, s, k∈Z,5r+ 15s-7
(b) sum of a rational and an irrational number is irrational.The formal representation of the given statement is:∀x∀r(x+r)/∈Q, wherex∈R-Q, r∈QBy contradiction, we assume that there is an irrational numberx /∈Qand arational numberr∈Qsuch thatr+x∈Q.