How can I determine the smallest quotient for all general cases?

Question

If i have a big number, like:

$39486432$ or $485921157$

And they ask me, with what number when dividing it, is it left with less residue? If you take into account the exact divisors. I would not like to use brute force, is there a wonderful theorem or approach to this?

Graphically: $\frac{485921157}{n}$, get the $n$ with less residue, without take into account the $0$ residue.

The smallest non-zero remainder is $1$. Do you really want this? Then the answer is trivial. — Dietrich Burde, Mar 22 '18 at 10:49
I suspect you would be better served by asking a more detailed problem. Little can be gathered by reading the Question in its current form. — hardmath, Mar 22 '18 at 15:06

score 1 · Accepted Answer · answered Mar 22 '18 at 10:42

1

Presumably a good answer, for a large number $n$, is "Divide it by $n-1$; that'll give a remainder of 1, which is as small as possible while not being zero."

Thus in one of your examples, we have $$ \frac{ 485921157}{ 485921156} = 1 \text {with a remainder of $1$}. $$

answered Mar 22 '18 at 10:42

John Hughes

93,729

1

More generally, (for $n > 1$) the integers which give a remainder of $1$ are the factors of $n - 1$. – Travis Willse Mar 22 '18 at 10:53
Good point, Travis --- thanks! – John Hughes Mar 22 '18 at 14:12

How can I determine the smallest quotient for all general cases?

1 Answers1