Final answer:
To find the remainder when p is divided by 4, we need to consider the given information. Statement (1) alone is not sufficient to answer the question, and statement (2) alone is not sufficient either. Together, the statements are still insufficient to answer the question.
Step-by-step explanation:
To find the remainder when p is divided by 4, we need to consider the given information. Let's analyze each statement:
(1) When p is divided by 8, the remainder is 5.
If p has a remainder of 5 when divided by 8, it means that p can be expressed as 8n + 5, where n is an integer. Since p is a positive odd integer, we can substitute 2k + 1 for n, where k is an integer. Therefore, p = 8(2k + 1) + 5 = 16k + 13.
Now, let's consider (2) p is the sum of the squares of two positive integers.
If p is the sum of the squares of two positive integers, it means p = a^2 + b^2, where a and b are positive integers.
Now, let's combine the two statements. Since p = 16k + 13 and p = a^2 + b^2, we can equate the two expressions and solve for k, a, and b. However, this approach does not allow us to determine the exact value of p or the remainder when p is divided by 4. Therefore, the statements (1) and (2) together are not sufficient to answer the question.
Hence, the answer is (E) - The two statements together are not sufficient.