Question

For nonnegative integers x and y, what is the remainder when x is divided by y?
1. \(\displaystyle \frac{x}{y}=13.8\)
2. The numbers \(\displaystyle x\) and \(\displaystyle y\) have a combined total of less than 5 digits.

• A. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
• B. Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
• C. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
• D. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• E. EACH statement ALONE is sufficient to answer the question asked

Hint:

When you see a decimal relationship like $\frac{x}{y}=13.8$, immediately convert it to a fraction in simplest form ($\frac{69}{5}$). This reveals the fundamental ratio between the integers x and y, which is often the key to solving the problem.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This data sufficiency question asks for the value of the remainder when integer \(x\) is divided by integer \(y\). The remainder is a specific value, so we need information that leads to a unique solution for \(x\) and \(y\), or at least for the remainder. The division algorithm states \(x = qy + r\), where \(q\) is the quotient and \(r\) is the remainder, with \(0 \le r < y\).

Step 2: Key Formula or Approach:
From the division algorithm, \(\frac{x}{y} = q + \frac{r}{y}\). The question asks for the value of \(r\).
The number of digits in an integer \(n\) is given by \(\lfloor \log_{10}(n) \rfloor + 1\).

Step 3: Detailed Explanation:
Analyzing Statement (1): \(\frac{x}{y} = 13.8\)
This equation can be rewritten as \(x = 13.8y\).
In the form of the division algorithm, this is \(x = 13y + 0.8y\).
Here, the quotient \(q\) is 13, and the remainder \(r\) is \(0.8y\).
To find the value of the remainder, we need the value of \(y\). Since \(y\) is unknown, the remainder cannot be determined.
For example:
- If \(y=5\), then \(x = 13.8 \times 5 = 69\). When 69 is divided by 5, the remainder is 4. (Note: \(0.8y = 0.8 \times 5 = 4\)).
- If \(y=10\), then \(x = 13.8 \times 10 = 138\). When 138 is divided by 10, the remainder is 8. (Note: \(0.8y = 0.8 \times 10 = 8\)).
Since the remainder can be different values, Statement (1) ALONE is not sufficient.
Analyzing Statement (2): The numbers x and y have a combined total of less than 5 digits.
Let \(D(n)\) be the number of digits in \(n\). The statement says \(D(x) + D(y) < 5\).
This information is too general. For example:
- Let \(x=100\) and \(y=3\). \(D(100)=3, D(3)=1\). Combined digits = 4 \(< 5\). Remainder of \(100 \div 3\) is 1.
- Let \(x=90\) and \(y=4\). \(D(90)=2, D(4)=1\). Combined digits = 3 \(< 5\). Remainder of \(90 \div 4\) is 2.
This statement alone is not sufficient.
Analyzing Statements (1) and (2) Together:
From Statement (1), we have \(\frac{x}{y} = 13.8 = \frac{138}{10} = \frac{69}{5}\).
Since \(x\) and \(y\) are integers, the ratio \(\frac{x}{y}\) must be \(\frac{69}{5}\) in its simplest form. This means \(x\) and \(y\) must be integer multiples of 69 and 5, respectively.
So, we can write \(x = 69k\) and \(y = 5k\) for some positive integer \(k\) (since x, y are non-negative, and y cannot be 0).
Now we use Statement (2): \(D(x) + D(y) < 5\).
Let's test values of \(k\):
- If \(k=1\): \(x = 69(1) = 69\), \(y = 5(1) = 5\). \(D(x) = D(69) = 2\). \(D(y) = D(5) = 1\). Combined digits = \(2+1 = 3\). Since \(3 < 5\), this is a valid solution.
The remainder when \(x=69\) is divided by \(y=5\) is 4.
- If \(k=2\): \(x = 69(2) = 138\), \(y = 5(2) = 10\). \(D(x) = D(138) = 3\). \(D(y) = D(10) = 2\). Combined digits = \(3+2 = 5\). Since 5 is not less than 5, this case is not valid.
- For any \(k > 1\), the number of digits will only increase, so their sum will be 5 or greater.
The only possible value for \(k\) is 1. This gives unique values for \(x\) and \(y\) as 69 and 5.
With unique values for \(x\) and \(y\), we can find a unique remainder.
Therefore, both statements TOGETHER are sufficient.

Step 4: Final Answer:
Neither statement alone is sufficient, but together they are sufficient to find a unique remainder. The correct option is (B).