For mathematical proofs, I want to be able to find all integers that divide another integer. For instance, integers that divide 50.
Following the definition that:
- $a$ and $b$ are integers where $b$ not equal to $0$
- $a$ is divisible by $b$ ($a/b$), if there is another integer $c$ where $a =
bc$
In the example above. I could say that 50 is divisible by 5 since $10\times5=50$ satisfying $a = bc$. But how can I prove it for all integers using the previous definition? Or am I looking at the wrong thing?
Thanks.
Best Answer
Can you use prime factorization? The best way to do this is say that $50 = 2^1 * 5^2$, and that any number that divides $50$ must be equal to $2^n * 5^m$, where $0 \leq n \leq 1$ and $0 \leq m \leq 2$. The full list is
$$ 2^0 * 5^0 = 1\\ 2^0 * 5^1 = 5\\ 2^0 * 5^2 = 25\\ 2^1 * 5^0 = 2\\ 2^1 * 5^1 = 10\\ 2^1 * 5^2 = 50 $$
If we have one of these numbers, $(2^n * 5^m)$, then we can show that $ (2^n * 5^m) * (2^{1-m} * 5^{2-m}) = 50$, so it divides $50$ by your definition of "divides."