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Please, before referring to another problem on the site or giving a link, I would like to say I read most of them such as: This, This, and many others but due to the (subtle) difference in my question, I'm having a really hard time to apply the methods used in other questions.

We yesterday had our first lecture about Number Theory. I have been trying to work on this problem since yesterday evening.

$$16a + 17b + 18c \equiv 19\pmod{100}$$ It's given that $ 1 \leq a,b,c \leq 99$ and that $a = 95\ and\ c=11$. We want to know what $b$ is.

I know that the answer is $53$ ( I used the naive way to get the answer) but I fail to do it according to the official methods. Can someone please explain this?

I tried this: ( i filled both a and c in)

$1718 +17b \equiv 19\ (mod\ 100)$

$17b \equiv 19-1718\ (mod\ 100)$

$17b \equiv -1699\ (mod\ 100)$

$$ x \equiv \frac{-1699}{17} \equiv \frac{-1699}{17} \equiv \frac{1}{17}\ i\ got\ stuck\ here$$

I also tried to work with it simplified: $$17b \equiv 1 (mod\ 100)$$ I thought i was almost done here since i was able to write it like this: $$b \equiv \dfrac{1}{17} (mod\ 100)$$ So i thought there is some $b$ number if i divide it by a $1/17$ i get $n$ as an answer and $100$ as rest but even doing so yielded in a wrong answer. Can someone please help?

  • See https://math.stackexchange.com/questions/25390/how-to-find-the-inverse-modulo-m – lab bhattacharjee Jan 12 '18 at 10:02
  • @labbhattacharjee I just applied the extended Euclidean algorithm such that $1s+100t = 1$ where t,s are integers. I found s = -99 , t =1. So that $1b \equiv 1 \pmod {100}$ but it was wrong for some reason –  Jan 12 '18 at 10:23
  • I don't think there are special ways for newbies, and no official ways. The (extended) Euclidean algorithm is just the most handy tool. –  Jan 12 '18 at 11:04
  • @ProfessorVector Could you please show me how The (extended) Euclidean algorithm will apply here. I tried and failed as i commented above –  Jan 12 '18 at 11:17
  • Sure. Extended Euclidean algorithm will give you the result ${1, {-47, 8}}$, meaning the GCD is $1$, and $(-47)\cdot17+8\cdot100=1$. And that means $1/17=-47=53\pmod {100}$. –  Jan 12 '18 at 11:43
  • @ProfessorVector I finally understand how the algorithm works and i was able to understand everything you did till " And that means ". May i ask how did you conclude that? –  Jan 12 '18 at 12:09

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HINT.-As a newbie you could act as follows:

First at all, note that $17$ is invertible modulo $100$ because it is not divisible by $2$ nor by $5$ so $17b \equiv 1 (mod\ 100)$ has solution.

Since $17n=17+17\cdots+17$ (n times), you could add successively $17$ plus $17$ until you get a number of the form $100x+1$ (this requires to be some patient having into account you know the answer is $53$).

Another shorter way is to add successively $100$ adding $1$ and dividing by $17$ until you have an exact quotient which is precisely your answer.This way requires just $9$ times instead of $53$.

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