We know that whatever positive integer we pick for n, we will arrive at the same result (otherwise they wouldn't have been able to ask the question this way).
So, let n be the smallest, easiest positive integer you can think of: 1.
So, we have 3^(8(1) + 3) or 3^11. When you have an exponent + remainder question, you should either recall or work out the cyclity of the units digit for that base (when you raise to consecutive-integer powers).:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
Look at the units digits. They went from 3, 9, 7, 1, and then back to 3 again. So, this pattern of 3, 9, 7, 1 will continue. That is, the 1st, 5th, 9th, 13th powers of 3 will all have units digit of 3 (the 2nd, 6th, 14th powers will all have units digit of 9, and so on).
We want the units digit of 3^11. There are four numbers in the cycle. 11/4 = 2R3. "R3" tells us which number in the cycle our units digit is. So, the units digits of 3^11 is the 3rd number in the cycle, or 7. Once we add the "2", our units digit will be 9. 9/5 leaves a remainder of 4, or choice E.
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