What is a number that is not prime and is not divisible by numbers 2 to 9? If a number is not divisible by numbers 2 to 9 can i say it's a prime number?
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1No. $143=11\cdot13$ is not prime and not divisible by any of these numbers. – saulspatz Mar 15 '20 at 19:26
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1Same with $121=11\cdot11$ – J. W. Tanner Mar 15 '20 at 19:27
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$11\times 13$ isn't prime and is not divisible by any number from $2$ to $9$. What do single digit numbers have so special? – Bernard Mar 15 '20 at 19:27
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I didn't think about multiplication between prime numbers, thank you for your answers. – Zetaku Mar 15 '20 at 19:30
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Remember the fundamental theorem of arithmetic. Every integer $n>1$ is either a prime number or can be written as a product of primes. There's nothing too special about the first few primes in particular.
Blue1.644934
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Generalizing your problem, imagine there is no composite number that's not divisible by the numbers 2 to n. This interval includes the primes $2,3,5,\dots,p$ (p is the last prime less than n) which means that any composite number can be divided by at least one of these primes. Now consider the number $(2\cdot 3 \cdot 5 \cdots p) +1$. It's clearly not divisible by any of the primes in the product. If it is a composite number, we proved our assumption wrong, and if it is a prime, it is necessarily a prime we didn't have before, just square it and we have a new composite number.
If primes are infinite then no interval can divide all composite numbers.
RicardoMM
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