1. Interpret the expression

The given expression is $3\times 7^{22} + 2\times 10^{22} - 4^4$

We need the remainder when this is divided by $18$.

So we must compute $3\cdot 7^{22} + 2\cdot 10^{22} - 4^4 \pmod{18}.$

2. Find $7^{22} \pmod{18}$

Notice: $7^2 = 49 \equiv 13 \pmod{18}$ Then, $7^3 = 7\cdot 7^2 \equiv 7\cdot 13 = 91 \equiv 1 \pmod{18}$

So powers of $7$ repeat with period $3$ modulo $18$: $7^3 \equiv 1 \pmod{18}$

Now, $22 = 3\cdot 7 + 1$ Hence, $7^{22} = (7^3)^7\cdot 7 \equiv 1^7\cdot 7 = 7 \pmod{18}$

Therefore, $3\times 7^{22} \equiv 3\times 7 = 21 \equiv 3 \pmod{18}.$

3. Find $10^{22} \pmod{18}$

First compute a few powers: $10^2 = 100 \equiv 10 \pmod{18}$ So once we reach $10^1$, the same value repeats: $10^n \equiv 10 \pmod{18} \quad \text{for all } n\ge 1$

Thus, $10^{22} \equiv 10 \pmod{18}$ Therefore, $2\times 10^{22} \equiv 2\times 10 = 20 \equiv 2 \pmod{18}.$

4. Find $4^4 \pmod{18}$

Compute directly: $4^2 = 16 \equiv -2 \pmod{18}$ $4^4 = (4^2)^2 \equiv (-2)^2 = 4 \pmod{18}$

So, $-4^4 \equiv -4 \pmod{18}.$

5. Combine all parts

Now add the congruences: $3\times 7^{22} + 2\times 10^{22} - 4^4 \equiv 3 + 2 - 4 = 1 \pmod{18}$

Therefore, the remainder is $\boxed{1}.$

6. Compare with stored answer

Stored correct answer: $15$

Our derived answer is $1$.

These do not match. The stored answer appears to be incorrect for the expression $3\times 7^{22} + 2\times 10^{22} - 4^4.$