Question

3. Identify the hypothesis and conclusion in the following statements. Then, find a counter example showing that the statement is false. Here n, m, and p are originally assumed to be integers.
Statement 1. If m and n are integers, then m/n is an integer.

Statement 2. If m and n are positive integers, then m - n is a positive integer.

Statement 3. If p is an odd prime number, then p^2 + 2 is a prime number.

Statement 4. If n - m is an even integer, then n and m are even integers.

Answer 1

Statement 1: This is true if n is a multiple of m then m/n is always an integer. Otherwise not possible. As for example let m=2 and n=3, then 2/3 is not a integer.

Statement 2:this is true only when m> n. Otherwise not possible. As for example, consider m=1,n=2 then m-n= - 1, which is a negative integer.

Statement 3: this is always false. Now we consider p=5, then 5^2+2=27, which is not a prime number because it's divisors are 1,3,9,27. But we know that a prime number has only two divisors namely 1,p(prime). So this statement is false.

Statement 4: this is false always. Now we consider n=5, m=3 then 5-3=2, a even number but 5,3 are not even. Hence it is false.