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Hello,

Problem 784
I'm having big trouble understanding problem 784, am I the only one ?

Let's call a pair of positive integers $p$, $q$ ($p \lt q$) <i>reciprocal</i>, if there is a positive integer $r\lt p$ such that $r$ equals both the inverse of $p$ modulo $q$ and the inverse of $q$ modulo $p$.

For example, $(3,5)$ is one reciprocal pair for $r=2$.

I really don't understand this sentence, can someone elaborate the meaning ?
In the literal sense, for me, 3 modulo 5 = 3 and 5 modulo 3 = 2, the inverse of these numbers is not 2.

Torpaz wrote: Tue Feb 08, 2022 9:37 pm Hello,

Problem 784
I'm having big trouble understanding problem 784, am I the only one ?

Let's call a pair of positive integers $p$, $q$ ($p \lt q$) <i>reciprocal</i>, if there is a positive integer $r\lt p$ such that $r$ equals both the inverse of $p$ modulo $q$ and the inverse of $q$ modulo $p$.

For example, $(3,5)$ is one reciprocal pair for $r=2$.

I really don't understand this sentence, can someone elaborate the meaning ?
In the literal sense, for me, 3 modulo 5 = 3 and 5 modulo 3 = 2, the inverse of these numbers is not 2.

You misunderstood the meaning of the word "inverse". It means the multiplicative inverse. In the example, it means that $2\equiv3^{-1} \bmod 5$, and $2\equiv5^{-1} \bmod 3$ are both true (or equivalently $2\cdot3\equiv1 \bmod 5$, and $2\cdot5\equiv1 \bmod 3$ are both true).

Oh ok that makes sense