Okay, so I need to find the units digit of 18 raised to the 6th power. Hmm, let me think. The units digit is the last digit of the number when it's written out. So, for example, if I have 18 squared, which is 324, the units digit is 4. But here, it's 18 to the 6th power. 

First, maybe I should recall that when dealing with units digits, only the units digit of the base affects the result. So, 18 has a units digit of 8. Therefore, the problem reduces to finding the units digit of 8^6. That simplifies things a bit. 

Now, I need to figure out the pattern of the units digits when 8 is raised to different powers. Let me list out the first few powers of 8 and see if there's a cycle or pattern in the units digits.

- 8^1 = 8. Units digit is 8.
- 8^2 = 64. Units digit is 4.
- 8^3 = 512. Units digit is 2.
- 8^4 = 4096. Units digit is 6.
- 8^5 = 32768. Units digit is 8.
- 8^6 = 262144. Units digit is 4.

Wait, so looking at this, the units digits cycle every 4 exponents: 8, 4, 2, 6, and then back to 8. So the cycle is 8, 4, 2, 6, repeating every 4 powers. 

Therefore, to find the units digit of 8^6, I can figure out where 6 falls in the cycle. Let's divide 6 by 4. 6 divided by 4 is 1 with a remainder of 2. So that means the cycle has completed one full cycle (4 exponents) and then 2 more exponents. 

Looking at the cycle: remainder 1 corresponds to the first position (8), remainder 2 corresponds to the second position (4). Since 6 divided by 4 is remainder 2, the units digit should be the same as 8^2, which is 4. 

Wait, but let me verify this with the actual calculation. 8^1 is 8, 8^2 is 64 (units digit 4), 8^3 is 512 (units digit 2), 8^4 is 4096 (units digit 6), 8^5 is 32768 (units digit 8), 8^6 is 262144 (units digit 4). Yep, that's consistent. 

So the pattern cycles every 4 exponents, and since 6 is 2 more than a multiple of 4, the units digit is the same as 8^2, which is 4. 

Alternatively, maybe I can use modular arithmetic here. The units digit is equivalent to the number modulo 10. So, 18^6 mod 10 is the same as (18 mod 10)^6 mod 10, which is 8^6 mod 10. 

Calculating 8^6 mod 10. Let's compute this step by step. 

First, compute 8^2 mod 10: 64 mod 10 = 4. 

Then, 8^4 = (8^2)^2 = 4^2 = 16 mod 10 = 6. 

Then, 8^6 = 8^4 * 8^2 = 6 * 4 = 24 mod 10 = 4. 

Same result. So that confirms it. 

Another way: using Euler's theorem. Since 8 and 10 are not coprime (they share a common factor of 2), Euler's theorem doesn't apply directly. But maybe we can break it down. 

Alternatively, since the cycle length is 4, we can use the property that 8^(4k + r) ≡ 8^r mod 10, where r is the remainder when the exponent is divided by 4. 

6 divided by 4 is 1 with remainder 2, so 8^6 ≡ 8^2 mod 10 ≡ 4 mod 10. 

Consistent again. 

Therefore, all methods lead to the conclusion that the units digit is 4. 

I think that's thorough. Let me just recap to make sure I didn't miss anything. The problem is asking for the units digit of 18^6. Since only the units digit of the base affects the result, we focus on 8^6. The units digits of powers of 8 cycle every 4 exponents: 8, 4, 2, 6. 6 divided by 4 is 1 with remainder 2, so the second digit in the cycle is 4. Therefore, the units digit is 4. 

Yes, that all checks out. I don't see any mistakes here. The answer should be 4.
Therefore, the units digit of $18^6$ is $\boxed{4}$.
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