To solve this problem, we need to prove two statements based on the given modular equivalences:

Given: $a_1 \equiv b_1 \pmod{m} \quad \text{and} \quad a_2 \equiv b_2 \pmod{m}$

Part (a): Show that $a_1 + a_2 \equiv b_1 + b_2 \pmod{m}$

Proof:

1. Since $a_1 \equiv b_1 \pmod{m}$, we know that $a_1 = b_1 + km$ for some integer $k$.
2. Similarly, since $a_2 \equiv b_2 \pmod{m}$, we have $a_2 = b_2 + lm$ for some integer $l$.
3. Adding these two congruences, we get: $a_1 + a_2 = (b_1 + km) + (b_2 + lm) = (b_1 + b_2) + (k + l)m$
4. This equation shows that $a_1 + a_2$ differs from $b_1 + b_2$ by a multiple of $m$.
5. Therefore, $a_1 + a_2 \equiv b_1 + b_2 \pmod{m}$, as required.

Part (b): Show that $a_1 a_2 \equiv b_1 b_2 \pmod{m}$

Proof:

1. Since $a_1 \equiv b_1 \pmod{m}$, we can write $a_1 = b_1 + km$ for some integer $k$.
2. Since $a_2 \equiv b_2 \pmod{m}$, we have $a_2 = b_2 + lm$ for some integer $l$.
3. Multiply $a_1$ and $a_2$: $a_1 a_2 = (b_1 + km)(b_2 + lm)$
4. Expanding this, we get: $a_1 a_2 = b_1 b_2 + b_1 lm + b_2 km + kl m^2$
5. Each term after $b_1 b_2$ is a multiple of $m$ (i.e., $b_1 lm$, $b_2 km$, and $kl m^2$ are all multiples of $m$).
6. Therefore, $a_1 a_2 \equiv b_1 b_2 \pmod{m}$, as required.

Thus, both parts (a) and (b) are proven.

Would you like further explanation on any part?

Further Questions:

1. What are the general properties of modular arithmetic in addition and multiplication?
2. Can we apply similar rules if we have more terms in the congruence relation?
3. How do these properties extend to powers, e.g., showing $a_1^n \equiv b_1^n \pmod{m}$?
4. How does modular arithmetic apply to practical cryptographic algorithms?
5. Could these proofs work with modular inverse operations?

Tip: In modular arithmetic, both addition and multiplication are closed operations. This closure property is helpful for constructing proofs similar to the one above.