You could just see what happens with 1 and 3 here (the only two different odd remainders when you divide by 4), to see that Statement 1 is sufficient. Or you could prove it algebraically: we can write an odd number as 2m + 1, so if a is the square of an odd integer, a = (2m + 1)^2 = 4m^2 + 4m + 1, and this number is 1 larger than a multiple of 4 (since 4m^2 + 4m is clearly divisible by 4), which is another way of saying "the remainder is 1 when we divide by 4". So Statement 1 is sufficient.
There is no relationship between the remainders you get dividing by 3 and the remainders you get dividing by 4, so Statement 2 is useless (testing any two consecutive multiples of 3 will demonstrate that), and the answer is A.
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