Three consecutive integers are selected from the integers 1 to 50, inc

Question Stem Analysis:

We choose three consecutive integers between 1 and 50, inclusive, and we need to determine the sum of the remainders when each integer is divided by x. No other information is provided in the question stem.

Statement One Alone:

\(\Rightarrow\) When the greatest of the consecutive integers is divided by x, the remainder is 0.

If x = 1, then the remainder when each of the integer is divided by x is 0. Thus, the answer is 0 + 0 + 0 = 0 in this case.

If x = 3, then the remainder when the middle integer is divided by x is 2, and the remainder when the smallest integer is divided by x is 1. Thus, the answer is 1 + 2 + 0 = 3 in this case.

Since there are more than one possible answers, statement one alone is not sufficient.

Eliminate answer choices A and D.

Statement Two Alone:

\(\Rightarrow\) When the least of the consecutive integers is divided by x, the remainder is 1.

If x = 3, then the remainder when the middle integer is divided by x is 2, and the remainder when the largest integer is divided by x is 0. In this case, the answer is 1 + 2 + 0 = 3.

If x > 3, then the remainder when the middle integer is divided by x is 2, and the remainder when the largest integer is divided by x is 3. In this case, the answer is 1 + 2 + 3 = 6.

Since there are more than one possible answers, statement two alone is not sufficient.

Eliminate answer choice B.

Statements One and Two Together:

If the the remainder when the smallest integer is divided by x is 1 and the remainder when the largest integer is divided by x is 0, then the remainder when the middle integer is divided by x can only equal 2. Thus, the sum of the three remainders is 1 + 2 + 0 = 3.

Statements one and two together are sufficient.

Answer: C