Proof by contradiction: x is prime number not equal to 3 => x/3 is not an integer

[zuby]

If x is a prime number not equal to 3, then x/3 is not an integer.
Step 1: Assume x/3 is an integer.
Step 2: x/3=n, x=6, x/3=2; x=9, x/3=3

I'm confused here:
if I assign 2 which is even, to x then output will be 0.67. it's still not an integer. is that prove wrong or I'm?

 

[ksdhart2]

To really utilize a proof by contradiction, your conclusion must contradict an assumption made previously (or a known mathematical fact) hence the "contradiction" in the name. We're told that x is some prime number, and it's not 3. We don't, however, know the exact value, but we don't actually need to. Your first step is correct, in that we assume the opposite of what we're trying to prove. Now, since we're assuming x/3 is an integer, let's call that unknown integer y. From that, what can you infer about x? Does this new piece of information contradict an assumption we made previously? Does it contradict a known mathematical fact? As a hint, think about what you know about prime numbers.

 

[pka]

If x is a prime number not equal to 3, then x/3 is not an integer.

Suppose that there is a prime number, \(\displaystyle P\ne 3\), such that \(\displaystyle \dfrac{P}{3}\in \mathbb{Z}\).
That means \(\displaystyle (\exists K\in\mathbb{Z}^+)[P=3N]\). By the supposition \(\displaystyle N\ne 1\). WHY?
Therefore, \(\displaystyle N>1\), which means that \(\displaystyle P\) cannot be a prime number. Explain why!
Now explain why this is a proof by contradiction.

 

[zuby]

To really utilize a proof by contradiction, your conclusion must contradict an assumption made previously (or a known mathematical fact) hence the "contradiction" in the name. We're told that x is some prime number, and it's not 3. We don't, however, know the exact value, but we don't actually need to. Your first step is correct, in that we assume the opposite of what we're trying to prove. Now, since we're assuming x/3 is an integer, let's call that unknown integer y. From that, what can you infer about x? Does this new piece of information contradict an assumption we made previously? Does it contradict a known mathematical fact? As a hint, think about what you know about prime numbers.

in 2nd step i contradicted the original statement...if x=6 then 6/3=2...except prime number when i use even number inplace of x like 2 or 4 it will also give non integer value.

 

[stapel]

in 2nd step i contradicted the original statement...if x=6 then 6/3=2...except prime number when i use even number inplace of x like 2 or 4 it will also give non integer value.

You assumed that the "if" was wrong. But if the "if" is not fulfilled, then the "then" never comes into play, so the "if-then" is not an issue.

As explained earlier, to prove an "if-then" "by contradiction", you must assume the "if" to be true (in this case, that x is prime) but the "then" to be false" (so x/3 is rational). Try doing it that way, following the hints and suggestions provided earlier. See where that leads.

 

[zuby]

To really utilize a proof by contradiction, your conclusion must contradict an assumption made previously (or a known mathematical fact) hence the "contradiction" in the name. We're told that x is some prime number, and it's not 3. We don't, however, know the exact value, but we don't actually need to. Your first step is correct, in that we assume the opposite of what we're trying to prove. Now, since we're assuming x/3 is an integer, let's call that unknown integer y. From that, what can you infer about x? Does this new piece of information contradict an assumption we made previously? Does it contradict a known mathematical fact? As a hint, think about what you know about prime numbers.

sorry 2 is prime but x=4, 4/3 = is irrational

 

[stapel]

sorry 2 is prime but x=4, 4/3 = is irrational

Yes, 2 is prime. No, 4 is not prime. (And it also has nothing to do with 2 being prime.) So assuming x to equal 4 is assuming the "if" to be wrong.

Instead, try using the information you've been provided, which specifically states that, to prove something "by contradiction", you must assume the "if" to be true. In other words, as mentioned previously, you must assume that x is prime! And you must assume that x is a generic prime, to represent all primes; you cannot assume x to equal any particular prime. You cannot assume x to be composite, let alone a specific composite.

Really: Please try doing the proof "by contradiction", rather than whatever other "method" you're trying to follow. In other words, start with "Let x be some prime number". Then assume "and let x/3 be rational". Then see where that leads.