Is there a formula to find every number divisible by $7$ but not $2,3$,or $5$? Not counting numbers but looking for specific solutions.

Question

All numbers divisible by 3 but not 2 can be found using

$6*n+3$

All numbers Divisible by 5 but not 3 or 2 can be found using

$10*n+5$

Is there a formula for divisible by 7 but not 5,3,or 2?

Answer 1

Let's look at your first case in more detail. All numbers which are not divisible by 2 are of the form $2k+1$. Multiply by 3 gives you all multiples of 3 which are not multiples of 2.

Next look at all numbers which are not multiples of 2 and 3. It's going to be periodic with period 6. So $6k+i$, where $i$ is an element of $(1,5)$. Multiply these two by 5 gives you $30k+5$ and $30k+25$.

The last case is similar, but $30k+i$, choosing $i$ so that it is not divisible by 2, 3 or 5. Then multiply these by 7.

Answer 2

The number must be of the form $N=7M$. Since $5,3,2$ are primes we need $(M,5)=(M,3)=(M,2)=1$, Let $M=10^na_n+a_{n-1}(10)^{n-1}+.....10a_1+a_0$; hence

$M$ not divisible by $5\iff a_0=1,2,3,4,6,7,8,9$

$M$ not divisible by $3\iff a_n +a_{n-1}+ .....+a_1+a_0\not\equiv3 (mod\space 3)$

$M$ not divisible by $2\iff M$ is odd.