Final answer:
The sum of the remainders is even if the sum of the units digits of the two original integers is 9.To find the sum of the remainders, divide the consecutive positive integers by 5 and add the remainders.
Step-by-step explanation:
To find the sum of the remainders, we need to divide two consecutive positive integers greater than 9 by 5 and add the remainders. Let's denote the first integer as x and the second integer as x+1. The remainder when x is divided by 5 is x%5, and the remainder when x+1 is divided by 5 is (x+1)%5. Therefore, the sum of the remainders is (x%5) + ((x+1)%5).
(1) The sum of the remainders is even: In this case, the possible values for x are 5, 10, 15, etc. The remainder when 5 is divided by 5 is 0, and the remainder when 6 is divided by 5 is 1. Therefore, the sum of the remainders is (0%5) + (1%5) = 1, which is odd. This statement is false.
(2) The sum of the units digits of the two original integers is 9: In this case, the possible values for x and x+1 could be 4 and 5, 9 and 10, 14 and 15, etc. The remainder when 4 is divided by 5 is 4, and the remainder when 5 is divided by 5 is 0. The sum of the remainders is (4%5) + (0%5) = 4, which is even. This statement is true.
Based on the above explanations, the correct answer is (2) The sum of the units digits of the two original integers is 9.