Find the least positive $x$ such that the expression $a + bx$ is divisible by $n$, where $a$, $b$, $n$ are assigned

Asked Apr 19 '21 at 11:53

Active Apr 19 '21 at 19:44

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I'm trying to solve this general problem(to solve the programming one).

We have assigned variables $a$, $b$ and $n$. What is the least positive $x$, so that $a + bx$ is divisible by $n$?

It is guaranteed that $x$ exists.

So far, I have found no relations between modulo or GCD. The main trouble are the independant variables.

edited Apr 19 '21 at 14:05

asked Apr 19 '21 at 11:53

Ver Nick

It is not always possible. Try $a=1, b=2, n=4$ – Henry Apr 19 '21 at 11:56
@Henry fixed it, thanks. $x$ is guaranteed to exist. – Ver Nick Apr 19 '21 at 11:58
is $x$ expected to be an integer? which independent variables are you referring to? – Siong Thye Goh Apr 19 '21 at 12:00
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A solution $x$ exists if and only if $a$ is divisible by $\operatorname{gcd}(b,n)$. – Geoffrey Trang Apr 19 '21 at 13:40
@SiongThyeGoh Yes, an integer. I'm referring to $a$, $b$ and $n$. – Ver Nick Apr 19 '21 at 14:06
See the linked dupes for methods to solve the congruence $, bx\equiv -a\pmod{n}\ \ $ – Bill Dubuque Apr 19 '21 at 19:45

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