As shown in image below, I understand Euler's Totient Function and all that except that one line marked in "red" namely, how does the inductive hypothesis prove that congruence involving Euler's totient function? Any clarification is appreciated. Thank you in advance.
It is strong induction on $n$. The author is saying:
Assume by strong induction that
$$x_{k}\equiv x_{k+1}\ (\text{mod }m)$$
for all $k\geq m$ and all $m<n$. The author is doing induction on $n$, not $k$. They then simply note that
$$\phi(b)\leq b-1\leq n-1<n$$
and thus satisfies the strong induction assumption that.