Final answer:
The remainder when a positive odd integer p, which leaves a remainder of 5 when divided by 8, is divided by 4 is always 1. This is due to the fact that 8k (where k is a whole number) is divisible by 4, and adding 5 to it will leave a remainder of 1 when divided by 4.
Step-by-step explanation:
If p is a positive odd integer and when p is divided by 8 the remainder is 5, we can determine the remainder when p is divided by 4. Since the remainder when p is divided by 8 is 5, this means that p can be expressed as 8k + 5, where k is a whole number. As such, p could be 5, 13, 21, 29, etc., all of which are odd numbers which are 5 more than a multiple of 8.
To determine the remainder when p is divided by 4, we can look at the series of numbers above and notice a pattern. Each of these numbers, when divided by 4, yields a remainder of 1. This is because 8k is divisible by 4, and thus the remainder is entirely determined by the 5, which leaves a remainder of 1 when divided by 4.
Therefore, it does not matter what the value of k is; the remainder when any number of the form 8k + 5 is divided by 4 is always 1. This is because 8k is always a multiple of 4, and adding 5 to a multiple of 4 will always leave a remainder of 1 when divided by 4.