Find the ones digit of $22^{22(11^{11})}$
First, observe that the units digit of $22^n$ is the same as the ones digit of $2^n$ for all positive integers $n$. Also, observe that the ones digits of $2^1, 2^2, 2^3, \ldots$ are 2, 4, 8, 6, 2, 4, 8, 6, .... Since $22(11^{11})$ is even but not divisible by 4, it leaves a remainder of 2 when divided by 4. Therefore, the units digit of $22^{22(11)^{11}}$ is the second digit in the repeating block 2, 4, 8, 6, which is $\boxed{4}$.
4
Level 3
Number Theory
Problem: Find the ones digit of $22^{22(11^{11})}$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the ones digit of a number, we can use modular arithmetic, specifically modulo 10. The ones digit of a number is the same as that number modulo 10. 

First, we need to determine the pattern of the ones digits of powers of 22. We can calculate the first few powers of 22:

- \(22^1 = 22\) (ones digit is 2)
- \(22^2 = 484\) (ones digit is 4)
- \(22^3 = 10648\) (ones digit is 8)
- \(22^4 = 234256\) (ones digit is 6)
- \(22^5 = 5153632\) (ones digit is 2)

From this, we observe a repeating cycle in the ones digits: 2, 4, 8, 6. The cycle has a length of 4.

Next, we need to determine which number in this cycle corresponds to \(22^{22(11^{11})}\). To do this, we need to find \(22(11^{11}) \mod 4\) because the cycle length is 4.

Calculating \(11^{11} \mod 4\):
- \(11 \equiv 3 \mod 4\)
- Therefore, \(11^{11} \equiv 3^{11} \mod 4\).

Now, we can find the powers of 3 modulo 4:
- \(3^1 \equiv 3 \mod 4\)
- \(3^2 \equiv 1 \mod 4\)

Since \(3^2 \equiv 1\), the powers of 3 modulo 4 will alternate:
- If the exponent is odd, \(3^n \equiv 3 \mod 4\).
- If the exponent is even, \(3^n \equiv 1 \mod 4\).

Since 11 is odd, we have:
- \(3^{11} \equiv 3 \mod 4\).

Thus, \(11^{11} \equiv 3 \mod 4\).

Now we calculate \(22(11^{11}) \mod 4\):
- \(22 \equiv 2 \mod 4\),
- Therefore, \(22(11^{11}) \equiv 2 \cdot 3 \equiv 6 \equiv 2 \mod 4\).

Now we know \(22(11^{11}) \equiv 2 \mod 4\). 

Referring back to the cycle of ones digits for powers of 22:
- \(22^1\) has a ones digit of 2,
- \(22^2\) has a ones digit of 4,
- \(22^3\) has a ones digit of 8,
- \(22^4\) has a ones digit of 6.

Since \(22(11^{11}) \equiv 2 \mod 4\), we take the second number in the cycle, which corresponds to \(22^2\).

Thus, the ones digit of \(22^{22(11^{11})}\) is the same as the ones digit of \(22^2\), which is 4.

The final answer is 4.<|END_OF_LATENT|> The final answer is $4$.