Question

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.

Answer 1

Answer:

a) $x = \sqrt{2}, \quad y = \sqrt{2}$

b)  n = 5

c) n = 6, a = 4, b = 3

Step-by-step explanation:

Incise a)

Let $x = \sqrt{2}, \quad y = \sqrt{2}$. Here $\sqrt{2}$ is a known irrational, and  $xy = \sqrt{2}\sqrt{2} = 2$ where the number 2 is not only rational but integer. 

Incise b)

If you take $n = 5$, 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.