Consider the statement, "If n is not prime, then n-2 is prime." What are some values of n that are counterexamples to this statement?

Answer 1

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6.

The first few prime numbers are: 2, 3, 5, 7, 11, 13, ...

The statement is: "If n is not prime, then n-2 is prime."

NOW, CONSIDER THE STATEMENT CAREFULLY !!

For these values of n, THIS STATEMENT IS TRUE !!

n = 4, 9, 15, 21, 25, 33, 38, 45, ...

Now, for what values is this statement FALSE [these are some counterexamples}:

n = 6, 8, 10, 12, 14, ... [note: ALL the even numbers]

Also, n= 27, 35, 51, 57, 65, ...

[note: these are composite numbers that are products of smaller prime numbers]

The statement, "If n is not prime, then n-2 is prime." is sometimes false and sometimes true.

Answer 2

Any even number greater than two is divisible by two, so it follows that no even number besides two is prime.

This means that any even value of n that is six or greater (8,10, 12, etc...) works as a counter example:

n - 2 when n = 6 → 6 - 2 = 4; 4 is divisible by 2 and thus not prime

n - 2 when n = 8 → 8 - 2 = 6; 6 is divisible by 2 and thus not prime.

n - 2 when n = 10 → 10 - 2 = 8; 8 is divisible by 2 and thus not prime.

Answer 3

Hello,

Let's start by making sure we understand what a prime number is. A prime number is a whole number greater than 1 whose only factors are 1 and itself.

Some examples of prime numbers are 2, 3, 5, 7, 11, 13, and 17. Note that the only numbers that can divide each of these numbers evenly are themselves and 1.

Now the question you're asking is to find counterexamples to the following statement: "If n is not prime, then n-2 is prime."

Let's take a number that is not prime and see if it can disprove this statement. 10, for example, is a number that has more factors than just itself and 1. The other two factors are 2, and 5.

If we take 10, and replace it with "n" in the statement, then the statement would read as: "If 10 is not prime, then 10-2 is not prime." This is false, because, 10 - 2 = 8, which is NOT prime. You could pretty much take any even whole positive number (except 4) to prove this statement false. Hope this helps!

Answer 4

There are infinite counterexamples of n that disprove this statement:

n=6 ⇒ n-2=4

n=8 ⇒ n-2=6

...

You can see for any even number n ≥ 6, none of which are prime, the even number n-2 ≥ 4 CANNOT be prime since they will all have a factor of 2.