If its not a trick question and you meant to put divided by, not "divisible by". The following is what we can conclude;
[(3^1)+2]/5= 1 + 0/5
[(3^2)+2]/5= 2 + 1/5
[(3^3)+2]/5= 5 + 4/5
[(3^4)+2]/5= 16 + 3/5
[(3^5)+2]/5= 49 + 0/5
[(3^6)+2]/5= 146 + 1/5
[(3^7)+2]/5= 437 + 4/5
[(3^8)+2]/5= 1312 + 3/5
[(3^9)+2]/5= 3937 + 0/5
Obviously from this we can see that there is a pattern of 0, 1, 4, 3. The question tells us that were going to be raising 3 to an odd degree becaues 8n+3 tell us so. N times and even number will make that even. Any even number plus an odd number, 8n+3, will be an odd number. As you can see from the evidence above there are different remainders for different odd values of n. When n=1, remainder is 0. When n=3, remainder equals 4. We can conlude that the, when devided by 5 the number will have a remainder of 0 or 4.
We know that there is a pattern and that the remainder must be 0 or 4. We need to think about what were raising it to and what those number fall on.
When n=1, 8*1+3= 11
n=2, 8*2+3= 19
n=3, 8*3+3= 27
If we devide these numbers by four the remainder will tell us where on the pattern of , 0,1,4,3 we will be at. If the remainder of the power divided by 4 is 1, then the remainder of the number in general will be 0. If the remainder of the power divided by 4 is 2, then the remainder of the number in general will be 1. If the remainder of the power divided by 4 is 3, then the remainder of the number in general will be 4.
So ,
11/4=2+3/4
19/4=4+3/4
27/4=6+3/4
Seeing this we can conclude that the power will always be in the third position in the pattern, thus the remainder of the number in general will always be 4.
