Neither statement alone is sufficient to determine whether n is a perfect square. Both statements are needed.
Statement 1:
A perfect square has an odd number of distinct factors. This is because the prime factorization of a perfect square consists of an even number of copies of each prime factor. For example, the prime factorization of 16 is 2^4, which has two copies of the prime factor 2. Therefore, 16 has a total of (2+1) = 3 distinct factors: 1, 2, and 16.
However, there are also positive integers that are not perfect squares that have an even number of distinct factors. For example, the prime factorization of 10 is 2*5, which has one copy of each prime factor 2 and 5. Therefore, 10 has a total of (1+1) = 2 distinct factors: 1 and 10.
Statement 2:
The sum of the distinct factors of a perfect square is always odd. This is because the sum of the prime factors of a perfect square is even, and the sum of an even number of odd numbers is always even.
However, there are also positive integers that are not perfect squares that have an even sum of distinct factors. For example, the prime factorization of 14 is 2*7, which has one copy of each prime factor 2 and 7. Therefore, 14 has a total of (1+1) = 2 distinct factors: 1 and 14, and the sum of these factors is 1+14 = 15, which is odd.