Analyzing Divisibility by 11: An In-Depth SEO Article

Understanding divisibility by 11 is essential in solving complex mathematical problems. This article delves into the concepts of divisibility and applies them to the problem of determining which expressions involving positive integers x and y are divisible by 11, given that 3x7y is a multiple of 11.

Introduction to Divisibility by 11

Divisibility by 11 can be determined by checking if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is divisible by 11. However, for mathematical expressions involving variables, we need a more structured approach. Let's explore the problem at hand.

Problem Statement

Given that 3x7y is a multiple of 11, we need to determine which of the following expressions A, B, C, and D are also divisible by 11.

Expression Analysis

Option A: 4x - 6y

Let's start by expressing 4x - 6y in terms of the given condition:

4x - 6y can be rewritten as:

4x - 6y 3x - 7y x - y

Given that 3x - 7y ≡ 0 (mod 11), we need to check if x - y can ensure divisibility by 11:

4x - 6y ≡ 0 (x - y) (mod 11) ≡ x - y (mod 11)

This does not guarantee that x - y is divisible by 11, as x - y can be any integer.

Option B: xy 4

Now let's analyze xy 4:

xy 4 ≡ x y 4 (mod 11)

We cannot directly relate x y to 3x - 7y, so we cannot conclude that this expression is divisible by 11.

Option C: 9x - 4y

Next, let's check 9x - 4y:

9x - 4y 33x - 7y - 23y ≡ 30 - 23y (mod 11)

This simplifies to:

9x - 4y ≡ -6y (mod 11)

Again, we cannot determine divisibility by 11 just from this expression.

Option D: 4x - 9y

Finally, let's check 4x - 9y:

4x - 9y 4x - 33y - 0 4x - 33y - 3x ≡ 4x - 33y - 3x (mod 11)

We can express 4x - 9y in terms of 3x - 7y:

4x - 9y 4x - 9y 4x - 7y - 2y ≡ 4x - 7y - 2y (mod 11)

Combining terms, we have:

4x - 9y ≡ 4x - 7y - 2y ≡ 4x - 7y - 20 ≡ (40 - 20) ≡ 0 (mod 11)

This expression simplifies to 0 modulo 11, indicating that 4x - 9y is divisible by 11.

Conclusion

After checking all options, we find that none of the expressions can be guaranteed to be divisible by 11 based on the given condition. However, if we specifically focus on 9x - 4y and 4x - 9y, we see that both can be expressed in terms of 3x - 7y but do not provide a clear divisibility. Therefore, while 4x - 9y simplifies in relation to 3x - 7y, it still requires additional information about x and y to conclusively determine divisibility.

Final Answer: None of the options are definitively divisible by 11 without additional constraints on x and y. However, 4x - 9y is the closest to ensuring divisibility based on the given condition.