I just want to confirm I am doing this problem correctly.
The problem asks to compute without a calculator:
$$
3 * \frac{2}{5} \pmod 7
$$
The way I am solving the problem:
$$
3 * \frac{2}{5} \bmod 7 \\
3 * 2 * \frac{1}{5} \bmod 7\\
\gcd(5,7) = 1 (\text{inverse exists})
$$
$$
(((3 \bmod 7) * (2 \bmod 7) * (1 \bmod 7)) \bmod 7) \\
(3 * 2 * 1) \bmod 7
$$
$$
6 \bmod 7 = 6
$$
Am I doing this correctly? Just started learning this in class. This is an even number practice problem out the book so I cannot check the answer lol.
Answer
We want to find $(3)(2)(x)\pmod{7}$, where $x$ is the inverse of $5$ modulo $7$, that is, where $5x\pmod 7=1$.
There are general procedures for finding inverses modulo $m$, but $7$ is a very small number, so we can do it efficiently by trial and error. Note that $(5)(3)$ has remainder $1$ on division by $7$. So $x\pmod 7=3$.
Thus we want to compute $(3)(2)(3)\pmod 7$. This is $4$.
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