Binomial Theorem Part 3

Concept:If $n$ is odd, $x^n + y^n$ is divisible by $x+y$.For the integral part of $(a + b\sqrt{c})^n$, use conjugate and binomial expansion.Explanation:Statement I: $25^{13} + 3^{13}$ is divisible by $25+3 = 28$.Since $28$ is divisible by $7$, $25^{13} + 3^{13} = 7I_1$, where $I_1$ is an integer.Similarly, $20^{13} + 8^{13}$ is divisible by $20+8 = 28$, so $20^{13} + 8^{13} = 7I_2$.Adding: $25^{13} + 20^{13} + 8^{13} + 3^{13} = 7(I_1+I_2)$. Hence divisible by $7$. Statement I is true.Statement II: Let $X = (7+4\sqrt{3})^{25}$, $I =$ integral part, $f =$ fractional part ($0 < f < 1$).Consider $(7-4\sqrt{3})^{25} = f'$, where $0 < f' < 1$ because $7-4\sqrt{3} \approx 0.07 < 1$.Using binomial expansion: $(7+4\sqrt{3})^{25} + (7-4\sqrt{3})^{25} = I + f + f'$ is an even integer (since it equals $2\sum \binom{25}{2k}7^{25-2k}(4\sqrt{3})^{2k}$).Since $0 < f+f'<2$ and $I$ integer, we must have $f+f'=1$. So $I+1$ is even, hence $I$ is odd. Statement II is true.Answer:Both Statement I and Statement II are true. Option C.