To test for divisibility, we use modular arithmetic.
A. Divisibility by 10:
7^1 = 7 (mod 10)
7^2 = 7*7 (mod 10) = 49 (mod 10) = 4*10+9 (mod 10) = 9 (mod 10)
7^3 = 9*7 (mod 10) = 63 (mod 10) = 6*10+3 (mod 10) = 3 (mod 10)
7^4 = 3*7 (mod 10) = 21 (mod 10) = 2*10+1 (mod 10) = 1 (mod 10)
Since this pattern keeps repeating, we have 7^82 (mod 10) = 7^[82 (mod 4)] (mod 10) = 7^2 (mod 10) = 9 (mod 10).
3^1 = 3 (mod 10)
3^2 = 3*3 (mod 10) = 9 (mod 10)
3^3 = 9*3 (mod 10) = 27 (mod 10) = 2*10+7 (mod 10) = 7 (mod 10)
3^4 = 7*3 (mod 10) = 21 (mod 10) = 2*10+1 (mod 10) = 1 (mod 10)
Since this pattern keeps repeating, we have 3^41 (mod 10) = 3^[41 (mod 4)] (mod 10) = 3^1 (mod 10) = 3 (mod 10).
Therefore, 7^82+3^41 (mod 10) = 9+3 (mod 10) = 12 (mod 10) = 2 (mod 10), so it is not divisible by 10.
B. Divisibility by 11:
7^1 = 7 = 7 (mod 11)
7^2 = 7*7 (mod 11) = 49 (mod 11) = 4*11+5 (mod 11) = 5 (mod 11)
7^3 = 5*7 (mod 11) = 35 (mod 11) = 3*11+2 (mod 11) = 2 (mod 11)
7^4 = 2*7 (mod 11) = 14 (mod 11) = 1*11+3 (mod 11) = 3 (mod 11)
7^5 = 3*7 (mod 11) = 21 (mod 11) = 1*11+10 (mod 11) = 10 (mod 11)
7^6 = 10*7 (mod 11) = 70 (mod 11) = 6*11+4 (mod 11) = 4 (mod 11)
7^7 = 4*7 (mod 11) = 28 (mod 11) = 2*11+6 (mod 11) = 6 (mod 11)
7^8 = 6*7 (mod 11) = 42 (mod 11) = 3*11+9 (mod 11) = 9 (mod 11)
7^9 = 9*7 (mod 11) = 63 (mod 11) = 5*11+8 (mod 11) = 8 (mod 11)
7^10 = 8*7 (mod 11) = 56 (mod 11) = 5*11+1 (mod 11) = 1 (mod 11)
Since this pattern keeps repeating, we have 7^82 (mod 11) = 7^[82 (mod 10)] (mod 11) = 7^2 (mod 11) = 5 (mod 11).
3^1 = 3 (mod 11)
3^2 = 3*3 (mod 11) = 9 (mod 11)
3^3 = 9*3 (mod 11) = 27 (mod 11) = 2*11+5 (mod 11) = 5 (mod 11)
3^4 = 5*3 (mod 11) = 15 (mod 11) = 1*11+4 (mod 11) = 4 (mod 11)
3^2 = 3*3 (mod 11) = 9 (mod 11)
3^3 = 9*3 (mod 11) = 27 (mod 11) = 2*11+5 (mod 11) = 5 (mod 11)
3^4 = 5*3 (mod 11) = 15 (mod 11) = 1*11+4 (mod 11) = 4 (mod 11)
3^5 = 4*3 (mod 11) = 12 (mod 11) = 1*11+1 (mod 11) = 1 (mod 11)
Since this pattern keeps repeating, we have 3^41 (mod 11) = 3^[41 (mod 5)] (mod 11) = 3^1 (mod 11) = 3 (mod 11).
Since this pattern keeps repeating, we have 3^41 (mod 11) = 3^[41 (mod 5)] (mod 11) = 3^1 (mod 11) = 3 (mod 11).
Therefore, 7^82+3^41 (mod 11) = 5+3 (mod 11) = 8 (mod 11) so it is not divisible by 10.
C. Divisibility by 4:
7^1 = 7 (mod 4) = 1*4+3 (mod 4) = 3 (mod 4)
7^2 = 3*7 (mod 4) = 21 (mod 4) = 5*4+1 (mod 4) = 1 (mod 4)
Since this pattern keeps repeating, we have 7^82 (mod 4) = 7^[82 (mod 2)] (mod 4) = 7^2 (mod 4) = 1 (mod 4).
3^1 = 3 (mod 4)
3^2 = 3*3 (mod 4) = 9 (mod 4) = 2*4+1 (mod 4) = 1 (mod 4)
Since this pattern keeps repeating, we have 3^41 (mod 4) = 3^[41 (mod 2)] (mod 4) = 3^1 (mod 4) = 3 (mod 4).
Therefore, 7^82+3^41 (mod 4) = 1+3 (mod 4) = 4 (mod 4) = 0 (mod 4), so it is divisible by 4, and the answer is C.
