Find the multiplicative order of $3$ modulo $301$.

Let $m$ and $n$ be two distinct positive integers. Find the minimal value of $(m+n)$ such that the last three digits of $2017^m$ and $2017^n$ are equal.

Find the multiplicative order of $17$ modulo $1000$.

Find all powers of $2$, such that after deleting its first digit, the new number is also a power of 2. For example, $32$ is such a number because $32=2^5$ and $2=2^1$.

An integer in the form of $F_n=2^{2^n}+1$ where integer $n\ge 1$ is called a Fermat's number. Let $d_n$ be any divisor of $F_n$. Show that $d_n\equiv 1\pmod{2^{n+1}}$.

Assume positive integer $n > 1$ satisfies $n\mid (2^n+1)$, prove $n$ is a multiple of $3$.

Let integers $l > m > n$ be the side lengths of a triangle satisfying $\left\{\frac{3^l}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$ where function $\{x\}$ returns the decimal part of real number $x$. Find the least possible value of this triangle's perimeter.