Division of whole numbers [closed]

Question

Is there any certain formula to figure out a problem like 9_8_7_ is divisible by 44 but not by 4? This is not a real question just similar to one I have seen. I realize that anything divisible by 44 will be divisible by 4, I do not need to find an answer to the problem I am trying to see if there is a faster way than trying all numbers in all situations! Thank you!

Answer 1

Every number divisible by $44$ is also divisible by $4$. To see this, consider a number $n$ divisible by $44$. This means that we can write $n$ as: $$ n = 44k$$ where $k$ is some integer. Now factor $44$ further, we get: $$ n = 4\times 11\times k = 2\times 2\times 11\times k. $$ Obviously, $n$ is divisible by $4$ (also by $2$, $11$, and $22$).

Answer 2

I think you are looking for "modular arithmetic." If you want to determine if a number is divisible by $n$, then you want to determine if that number is congruent to $0$ mod $n$. Methods for doing this depend on $n$. Your question is far to general to have any sort of formula or specific trick.

In fact, it is impossible to tell if $4$ divides a number of the form 9_8_7_ without knowing the last digit, but it will be divisible by $4$ iff the last digit is $2$ or $6$. This is because $4$ divides $20$ and in turn $100$, so first of all we only need to look at the last two digits, and secondly we only need to consider that last two digits mod $20$. Specifically, 9_8_7x is congruent to $1x$ mod 20, and the only numbers divisible by $4$ in the teens are $12$ and $16$.