Statement 1 = Sum of the factors of integers C is ODD.
Factors of 9 = 1, 3, 9, sum = ODD.
Factors of 8 = 1, 2, 4, 8, sum = ODD.
Hence, Statement 1 is not SUFFICIENT.

Statement 2 = Number of distinct factors of integer C is ODD.
For an integer $$N=x^ay^{b\ }\ $$ the number of factors
$$=\left(a+1\right)\cdot\left(b+1\right)$$
if $$\left(a+1\right),\left(b+1\right)$$ will also be ODD or A, B are even. Hence we have
$$N=x^{even}\cdot y^{even}$$
N=square of an integer / numbers
Statement 2 is SUFFICIENT.
Final Answer = Option B because statement 2 alone is SUFFICIENT.