Confusion about proving a statement involving modulo arithmetic

Question

I'm trying to prove that if $a \equiv b \pmod{4}$, then $123^a \equiv 33^b \pmod{10}$.

What I have so far is this.

First, applying the definition of the modulus function, it follows that $a \equiv n \pmod{4} \iff a = 4k + b$. Therefore,

$$ \begin{align} 123^a &\equiv 123^{4k+b} \pmod{10}\\ &\equiv 3^{4k+b} \tag{confused} \\ &\equiv \left(3^{4}\right)^k \cdot 3^{b} \\ &\equiv (1)^k \cdot 3^b \\ &\equiv 3^b \end{align} $$

Then as $33^b \equiv 3^b \pmod{10}$, by the transitivity of the modulo operator, $123^a \equiv 33^b \pmod{10}$ (assuming that $a \equiv b \pmod{4})$.

However, I'm not sure why it's possible to substitute values inside the modulo expression. For example, I tried to prove that

$$ \left(3^4\right)^k \equiv 1 \pmod{10} $$

Applying the definition

$$ \left(3^{4}\right)^k \equiv 3^4 \cdot 3^4 \cdot ... \cdot 3^4 $$

But from here I got stuck. I think I want to use the fact that if $a \equiv b \pmod p$ and $c \equiv d \pmod{p}$, then $ac \equiv bd \pmod{p}$, so if $a = c = 3^4$, then $b = d = 1$ satisfy the antecedent, so then the consequent is also true (that $3^k \cdot 3^k \equiv 1 \cdot 1 \equiv 1 \pmod{10}$).

But I'm not sure how to solve this in general (i.e. that in any case it's always possible to substitute equivalent values into equations modulo $m$, if that even is true, which I'm not sure).

Sorry, I find modulo arithmetic really confusing so any help is much appreciated.

Answer 1

What you're trying to do is fine. If you want to look at it rigorously, then you can do a proof via induction to show that $x \equiv y \mod m \implies x^n \equiv y^n \mod m$ for positive integers $a$:

First, $x \equiv y \implies x^1 \equiv y^1$ is obvious.

Then, assuming that $x \equiv y \mod m \implies x^k \equiv y^k \mod m$, we have:

$\begin{eqnarray}x \equiv y \mod m \implies \\ x^{k+1} & = & x^k \times x \\ & \equiv & y^k \times x \mod m & \mbox{by assumption} \\ & \equiv & y^k \times y \mod m \\ & \equiv & y^{k+1} \mod m \end{eqnarray}$

which completes the induction step.

So you can definitely say that $(33)^4 \equiv 3^4 \mod 10$, although the reverse won't always be true (i.e. it isn't necessarily true that $x^n \equiv y^n \implies x \equiv y$).