The answer is actually 1, and not 0, and here's why.
Write 3^36 into (3^3)^12. Note that the exponents 3 and 12 multiply to 36
Now focus on 3^3 which turns into 27. The last digit here is 7. It turns out that this is the only thing that matters when it comes to finding the final answer.
So instead of 27^12, we can look at 7^12
Break 7^12 into (7^3)^4, then note how 7^3 = 343. Again the units digit is all we care about. So instead of 343^4, we just look at 3^4
3^4 = 81 and the units digit here is 1 which is the final answer.
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Here's an alternative approach using modular arithmetic. This is the set of tools that helps us find remainders. If we work with mod 10, then we can find the units digit of any complicated expression such as 3^36
3^36 = (3^3)^12 (mod 10)
3^36 = 27^12 (mod 10)
3^36 = 7^12 (mod 10)
3^36 = (7^3)^4 (mod 10)
3^36 = 343^4 (mod 10)
3^36 = 3^4 (mod 10)
3^36 = 81 (mod 10)
3^36 = 1 (mod 10)
We effectively follow the same steps as mentioned above, but we're using a more formal set of steps to show.