Prove: if x^2 is divisible by 3 then x is divisible by 3
Thanks so much anybody who can help me on this.
Fundamental Theorem of Arithmetic -- from Wolfram MathWorld
... in normal practice x indicates a real variable, for integer numers usually letters like n, m, i, j or k are used...
Kind regards
If is divisible by 3, then its prime factorization contains at least one 3. Since 3 is a prime number, it follows that x must also have a factor of 3.
(Since the question is about "divisibility", it is clear that x must be an integer so using "x" rather than "n" doesn't bother me.)
You want to prove: If is divisible by 3 then is divisible by 3.
The contrapositive of this is: If is not divisible by 3 then is not divisible by 3.
To prove the contrapositive, note that any integer can be written as or , where is some other integer. Since we have said that is not divisible by , then we can let or .
Case 1:
which is not divisible by 3.
Case 2:
which is not divisible by 3.
Since we have shown that if is not divisible by 3, then is not divisible by 3, the contrapositive is also true.
You are more than welcome. Just one thing though, you should note that proving a statement by proving the contrapositive is not considered a direct proof, but rather an indirect proof. However, I think it's important to see how often it is easier to prove something indirectly than directly.