Question

If the product of two integers x and y is less than 82 with y being a multiple of three. What is the highest value that x may have?

• A. 13
• B. 42
• C. 27
• D. 30
• E. 34

Hint:

When trying to maximize one variable in an inequality like \(x \cdot y<K\), if the variables are positive, you should minimize the other variable. Always check the constraints (e.g., integer, multiple of three) to find the appropriate minimum value.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We are given an inequality involving the product of two integers, x and y. We are also given a condition on y. We need to find the maximum possible integer value for x.
Step 2: Key Formula or Approach:
The problem states:
1. \(x, y\) are integers.
2. \(x \cdot y<82\).
3. \(y\) is a multiple of three.
To maximize the value of x, we should make the other factor, y, as small as possible, while still being positive (to avoid flipping the inequality sign).
Step 3: Detailed Explanation:
The condition is \(x \cdot y<82\). To find the highest possible value for x, we need to analyze the values y can take.
y is an integer and a multiple of three. Possible values for y are \(..., -6, -3, 3, 6, 9, ...\).
If we choose a negative value for y (e.g., y = -3), the inequality becomes \(x \cdot (-3)<82\), which simplifies to \(-3x<82 \implies x>-82/3 \implies x>-27.33\). This puts a lower bound on x, but not an upper bound, so x could be infinitely large. The phrase "highest value" implies we are looking for a maximum, so we should assume x and y are positive.
Let's assume x and y are positive integers. To maximize x in the inequality \(x<\frac{82}{y}\), we must choose the smallest possible positive value for y.
The smallest positive integer that is a multiple of three is \(y = 3\).
Substituting \(y=3\) into the inequality:
\[ x \cdot 3<82 \] Now, solve for x:
\[ x<\frac{82}{3} \] \[ x<27.33... \] Since x must be an integer, the highest integer value that x can take while being less than 27.33... is 27.
Step 4: Final Answer:
The highest value that x may have is 27.