Given $a\le1025$ , Find Minimal non-negative $y$ such that $1025 \equiv 0\mod (a+y)$

Question

General solution that works with not only $1025$ but also any given number is preferable.

Thanks for the answer @Peter , although it is correct this is not the minimal y. — elise, Aug 27 '17 at 17:47
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. — Math Lover, Aug 27 '17 at 17:50

score 2 · Answer 1 · answered Aug 27 '17 at 17:41

2

You need to find the smallest factor of $1025$ greater than or equal to $a$.

$1025 = 5^2\cdot41$ so only has six factors: $\{1,5,25,41,205,1025\}$. Choose $y$ to make $a{+}y$ equal to one of these.

answered Aug 27 '17 at 17:41

Joffan

Great answer, thanks! I managed to get this one but was hopping to find a "prettier" one. – elise Aug 27 '17 at 17:50
@elise In my opinion this is a pretty answer, I believe Joffan was trying to keep his response simple so that it would be easy to understand. What this says for the general case is that, for $b \geq (a+y)$ we have $b\equiv 0 \mod (a+y)$ if and only if $(a+y)$ divides $b$. i.e. there exists $k$ such that $k \cdot (a+y) = (b)$ and thus $(a+y)$ is the smallest factor of $b$ greater than or equal to $a$. – mm8511 Aug 27 '17 at 18:17

1 Answers1