Proof & Structures: Showing n≤0 for Prime/Composite Number

In summary, the conversation discusses two questions involving the quadratic expression n^2 - 14n + 40. The first question asks to show that if n ≤ 0, then the expression is a composite number. The second question asks to determine all integer values of n for which the expression is prime, with a hint to factorise the expression and consider possible values of n.
  • #1
johnny009
7
0
Hi There,

My apologies, there was an error...in a previous question, which I POSTED ....last week.

This question has now been withdrawn, & replaced with the following :

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a) Show that if n ≤ 0, then n^2 - 14n + 40 is a composite number.

b) Determine, with explanation, all integer values of 'n' for which ...n^2 -14n + 40 is prime

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any guidance on these issues.....will be fantastic!

Thanks a lot.

John
 
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  • #2
johnny009 said:
a) show that if n ≤ 0, then n^2 - 14n + 40 is a composite number.

b) Determine, with explanation, all integer values of 'n' for which ...n^2 -14n + 40 is prime
Hint: Factorise the quadratic expression $n^2 - 14n + 40.$
 
  • #3
If you have factorised $n^2 - 14n + 40$, then the only cases where this could represent a prime would be when one of the factors is $\pm1$ and the other factor is (plus or minus) a prime. Which values of $n$ give rise to those possibilities?
 

1. How do you prove that a number is prime or composite?

To prove that a number is prime, you need to show that it is only divisible by 1 and itself. You can do this by trying to divide the number by all numbers between 2 and the square root of the number. If none of these divisions result in a whole number, then the number is prime. On the other hand, to prove that a number is composite, you need to find at least one number other than 1 and itself that can divide into the number evenly.

2. What is the significance of showing n≤0 for prime/composite numbers?

Showing that n≤0 for prime numbers is significant because it confirms that the number is not divisible by any number other than 1 and itself, which is the definition of a prime number. Similarly, showing that n≤0 for composite numbers proves that the number has at least one factor other than 1 and itself.

3. Can you use any number for n when proving primality or compositeness?

Yes, you can use any number for n when proving primality or compositeness. However, it is recommended to use numbers that are easy to work with and have a known prime or composite status. For example, using 2 or 3 as n is often a good starting point.

4. Are there any shortcuts or tricks for proving a number is prime or composite?

There are some techniques and shortcuts that can help in proving a number is prime or composite, such as the Sieve of Eratosthenes or the divisibility rules. However, these methods are not foolproof and may not work for every number. In general, proving primality or compositeness requires careful analysis and testing of potential factors.

5. Why is it important to know whether a number is prime or composite?

Knowing whether a number is prime or composite is important in many areas of mathematics, including number theory, cryptography, and computing. Prime numbers, in particular, have many applications and are the building blocks of many mathematical concepts. Additionally, identifying prime and composite numbers can help in solving mathematical problems and making predictions in fields such as economics and biology.

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