Let $X = \{1, 2, 3, 4, 5, 6, 7\}.$ For every $n \in X$, write $n^2 - 3n^5 = 7q_n + r_n, 1 \leq r_n \leq 7.$ Define a function $f: X \to X$ by $f(n) = r_n.$
(a) Find an element $\alpha \in S_7$ that can be used to represent this bijection $f$.
(b) Determine $sgn(\alpha)$.
My question is how can I determine $\alpha$ when I don't know what $q_n$ is. I have no idea at all how to solve this question.
Hint: For example, when $n=4$, you get $4^2 - 3\cdot 4^5 = -3056 = 7\cdot(-437) + 3$, so that $r_4 = 3$. So $f(4) = 3$. If you do this for each element of $X$, you should be able to figure out what the permutation is.