Q:

Give a counterexample to each claim:(a) If x, y are both irrational then xy is also irrational(b) For all n ? N, 2n 2 + 5 is prime(c) For all a, b, n ? N, if n | ab then n | a or n | b.

Accepted Solution

A:
Answer:a) [tex]x = \sqrt{2}, \quad y = \sqrt{2}[/tex]b)  n = 5c) n = 6, a = 4, b = 3Step-by-step explanation:Incise a)Let [tex]x = \sqrt{2}, \quad y = \sqrt{2}[/tex]. Here [tex]\sqrt{2}[/tex] is a known irrational, and  [tex]xy = \sqrt{2}\sqrt{2} = 2[/tex] where the number 2 is not only rational but integer. Incise b)If you take [tex]n = 5[/tex], you will get 2(25) + 5 = 55 that is not prime, because 5 divides 55.Incise c)Here let n = 6, a = 4, b = 3. We can see that ab = 12, and of course 6 divides 12 (n | ab). But, also 6 does not divides 4 and  6 does not divides 3.