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wolverine [178]
2 years ago
8

Prove that the natural number x is prime if and only if x>1 and there is no positive integer greater than 1 and less than or

equal to \sqrt{x} that divides x.
Mathematics
1 answer:
UNO [17]2 years ago
3 0

There are several ways to prove a mathematical statement. These ways include: by contradiction, by induction, contraposition, etc

<h3>How to prove the statement by contradiction</h3>

To prove the statement, we make use of proof by contradiction

From the question:

x is a composite number such that: x > 1

This means that 1 < a < x is a factor of x

This can be represented as:

x = a * b

Where:

  • a  and b  are  positive  integers
  • 1 < a, b < x

Assume that b is less than or equal to a.

Also, let

b > \sqrt x​

This means that:

\sqrt x < b \ge a

The above becomes

\sqrt x < a

Rewrite as:

a > \sqrt x

So, we have:

x = a * b > \sqrt x * \sqrt x = x

The above means that:

x > x

The above inequality is a contradiction because, a number x cannot be greater than itself

This means that, the supposition is wrong.

Hence, the given statement has been proved by contradiction

Read more about proof by contradiction at:

brainly.com/question/8062770

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Which statements are true?<br>(look at picture)​
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Answer:

D is true, E is true, F is true

Step-by-step explanation:

Hello,

n is an odd number it means that n = 2a+1 where a is integer

A

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this is <u>not</u> an odd number

B

n(n-1)=(2a+1)2a

this is <u>not</u> an odd number

C

(n-1)^2=(2a+1-1)^2=4a^2

this is <u>not</u> an odd number

D

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so this is an odd number

E

n(n-2)=(2a+1)(2a-1)=4a^2-1=4a^2-2+1=2(2a^2-1)+1

so this is an odd number

F

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hope this helps

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