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I'm Computer Science student. Last day, my teacher say this to the class room: "If a number is not divisible by 2, 3, and 5, mean that number is prime. This because odd numbers are divisible by 3 or 5, and even numbers by 2".

That can be true? If it's or not, why?

Edit: I'm not an mathemathician. I necer belive the words of my teacher. I just want an professional clarify.

Moige
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    I'm voting to close this question because I think it's one you should have been able to answer for yourself with a little thought about what "prime" means and a little arithmetic on numbers less than $100$. Did your teacher really say this seriously? – Ethan Bolker Aug 04 '18 at 15:14
  • I swear. And make a little function in Python to "prove" this. 3 simple conditionals: if n mod {3, 2, 5} is not 0, then is prime. – Moige Aug 04 '18 at 15:20
  • You may be interested to read about the Sieve of Eratosthenes, a (tedious) way of finding all prime numbers within a certain range. The claim that all numbers not divisible by $2,3,5$ are prime should be obviously false. Those numbers which are not divisible by $2,3,5$ are simply those numbers still left in the sieve after the first three iterations of the algorithm, but there are still many more steps left to perform in the algorithm (getting rid of all proper multiples of $7$, all proper multiples of $11$, etc...). – JMoravitz Aug 04 '18 at 15:20
  • For small counterexamples of numbers which are divisible by none of $2,3,5$ but which are not prime, consider $7\times 7 = 49, 7\times 11 = 77, 7\times 13 = 91, 11\times 11 = 121,\dots$ – JMoravitz Aug 04 '18 at 15:22
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    I wonder if this is a serious question. I don't believe that you really think any odd number is divisible by 3 or 5. I'm sure you know that $49=7^2$. – Mark Aug 04 '18 at 15:22
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    I would be interested in the sentence or two that your teacher said before or after those sentences. Perhaps you missed words like this: "Is the following true?" Even if not, a much, much better alternative (than coming to this web site) would have been to raise your hand and say "That is not true". – Lee Mosher Aug 04 '18 at 15:25
  • A correction to the statement that will make this true: "If $x$ is a natural number between $2$ and $48$ and $x$ is not divisible by any of $2,3,5$ then $x$ is prime." Once the upper bound on $x$ is removed however, this as mentioned is false. We also needed the lower bound since $1$ is not considered a prime number. (Note: $x$ is a composite number if and only if there is some prime $p$ which divides $x$ such that $p\leq \lfloor \sqrt{x}\rfloor$, i.e. when searching for divisors, we need not look any further than the square root.) – JMoravitz Aug 04 '18 at 15:29
  • @Mark I never belive that. I always think that need to do a brute force algorithm to check all numbers above the possible prime. I only wanted to a real mathemathician clarify me. – Moige Aug 04 '18 at 15:31
  • And I just get -3 point. Ok. Thanks to all that response me with fundament. – Moige Aug 04 '18 at 15:37

1 Answers1

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$49$ is not divisible by any of those number but it is not a prime.

In general, take a prime number that is not $2,3,$ or $5$, consider $p^n$ where $n \ge 2$ works as a counter example.

Siong Thye Goh
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