# How to solve this equation for d [closed]

Solve 17d mod 24 = 1. Would it be d = 17 inverse mod 24 and then solved using EEA?

## closed as off-topic by Shailesh, Michael Hoppe, Lord Shark the Unknown, Leucippus, user10354138Nov 13 '18 at 6:44

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Yes that means $$17d = 24k +1$$ or as you put it , $$d$$ is the multiplicative inverse of $$17$$ mod $$(24).$$

Yes, you have it correct. Since $$17d\equiv 1\;(mod\;24)$$, there exists an integer $$k$$ such that $$17d-1=24k$$. This is a linear Diophantine equation, and since $$gcd(17,24)=1$$, there is a guarenteed solution. In this particular case, we have a solution of $$d=17$$ and $$k=12$$, which can be found using the Euclidean Algorithm.

the Extended Euclidean algorithm applied to $$17$$ and $$24$$ will give us $$k,m \in \mathbb{Z}$$ such that

$$17k + 24m = \gcd(17,24)=1$$

and taking this equation modulo $$24$$ we get

$$17k \equiv 1 \pmod{24}$$

showing that $$k$$ is indeed the inverse of $$17$$ modulo $$24$$ (and $$17$$ of $$k$$ as well).

I get $$5 \cdot 24 + (-7)\cdot 17 = 1$$ with EEA.