If x and y are positive integers such that y is a multiple of 5 and 3x + 4y =...

1. Translate the problem requirements: We need to find what number x must be divisible by, given that y is a multiple of 5 and both x and y are positive integers satisfying \(3\mathrm{x} + 4\mathrm{y} = 200\)
2. Express x in terms of y using the constraint equation: Isolate x from the equation \(3\mathrm{x} + 4\mathrm{y} = 200\) to see what determines x's value
3. Apply the multiple of 5 condition to analyze x's structure: Since y is a multiple of 5, substitute \(\mathrm{y} = 5\mathrm{k}\) and examine what this means for x
4. Determine divisibility requirements for x: Analyze the resulting expression for x to identify what number it must be divisible by

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're given and what we need to find.

We have two positive integers, x and y, with these conditions:
• y is a multiple of 5 (meaning y could be 5, 10, 15, 20, etc.)
• \(3\mathrm{x} + 4\mathrm{y} = 200\) (this is our constraint equation)
• We need to find what number x must be divisible by

Think of this like a puzzle where we have a specific relationship between x and y, and we need to figure out what pattern x must follow.

Process Skill: TRANSLATE

2. Express x in terms of y using the constraint equation

Let's solve for x from our equation \(3\mathrm{x} + 4\mathrm{y} = 200\).

Starting with: \(3\mathrm{x} + 4\mathrm{y} = 200\)
Subtract \(4\mathrm{y}\) from both sides: \(3\mathrm{x} = 200 - 4\mathrm{y}\)
Divide both sides by 3: \(\mathrm{x} = \frac{200 - 4\mathrm{y}}{3}\)

Now we can see that x depends on what y is. Since we know y has to be a multiple of 5, let's use that information.

3. Apply the multiple of 5 condition to analyze x's structure

Since y is a multiple of 5, we can write \(\mathrm{y} = 5\mathrm{k}\) where k is any positive integer.

Substituting this into our expression for x:
\(\mathrm{x} = \frac{200 - 4\mathrm{y}}{3}\)
\(\mathrm{x} = \frac{200 - 4(5\mathrm{k})}{3}\)
\(\mathrm{x} = \frac{200 - 20\mathrm{k}}{3}\)

Let's factor this expression:
\(\mathrm{x} = \frac{200 - 20\mathrm{k}}{3}\)
\(\mathrm{x} = \frac{20(10 - \mathrm{k})}{3}\)

For x to be a positive integer, \((10 - \mathrm{k})\) must be divisible by 3. But more importantly, let's see what this tells us about x's divisibility.

Process Skill: MANIPULATE

4. Determine divisibility requirements for x

Looking at our expression \(\mathrm{x} = \frac{20(10 - \mathrm{k})}{3}\), we can rewrite this as:
\(\mathrm{x} = \frac{20}{3} \times (10 - \mathrm{k})\)

For x to be an integer, \((10 - \mathrm{k})\) must be divisible by 3. Let's call \(\frac{(10 - \mathrm{k})}{3} = \mathrm{m}\), where m is an integer.

Then: \(\mathrm{x} = 20\mathrm{m}\)

This means x must always be a multiple of 20!

Since \(20 = 4 \times 5 = 2^2 \times 5\), any multiple of 20 is automatically a multiple of:
• 1, 2, 4, 5, 10, and 20

Looking at our answer choices:
A) 3 - Not necessarily (20 is not divisible by 3)
B) 6 - Not necessarily (20 is not divisible by 6)
C) 7 - Not necessarily (20 is not divisible by 7)
D) 8 - Not necessarily (20 is not divisible by 8)
E) 10 - Yes! Every multiple of 20 is also a multiple of 10

Process Skill: APPLY CONSTRAINTS

5. Final Answer

Since \(\mathrm{x} = 20\mathrm{m}\) for some integer m, x must be a multiple of 20, which means x must be a multiple of 10.

Let's verify with an example: If \(\mathrm{k} = 1\), then \(\mathrm{y} = 5\) and \(\mathrm{x} = \frac{20(10-1)}{3} = 60\). Check: \(3(60) + 4(5) = 180 + 20 = 200\) ✓
If \(\mathrm{k} = 4\), then \(\mathrm{y} = 20\) and \(\mathrm{x} = \frac{20(10-4)}{3} = 40\). Check: \(3(40) + 4(20) = 120 + 80 = 200\) ✓

Both 60 and 40 are multiples of 10.

The answer is E) 10.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "multiple of 5" means for y

Students might think y can only be 5, rather than understanding that y can be any multiple of 5 (like 5, 10, 15, 20, etc.). This leads them to solve only for the specific case where y = 5, missing the general pattern that x must follow.

2. Not recognizing the need to express constraints systematically

Students may try to test random values or use trial-and-error instead of systematically expressing y as 5k and substituting into the equation. This approach becomes inefficient and may not reveal the underlying divisibility pattern for x.

3. Focusing on finding specific values rather than general divisibility

Since the question asks "x must be a multiple of which of the following," students might get distracted trying to find all possible values of x instead of focusing on what x is always divisible by regardless of the specific values.

Errors while executing the approach

1. Algebraic manipulation errors when solving for x

When rearranging \(3\mathrm{x} + 4\mathrm{y} = 200\) to get \(\mathrm{x} = \frac{200 - 4\mathrm{y}}{3}\), students might make sign errors or incorrect fraction operations, leading to wrong expressions for x in terms of y.

2. Incorrect substitution of y = 5k

When substituting \(\mathrm{y} = 5\mathrm{k}\) into \(\mathrm{x} = \frac{200 - 4\mathrm{y}}{3}\), students might incorrectly compute \(4(5\mathrm{k})\) or make errors in simplifying \(\frac{200 - 20\mathrm{k}}{3}\), missing the factorization that leads to \(\mathrm{x} = \frac{20(10-\mathrm{k})}{3}\).

3. Not properly analyzing the divisibility condition

Students might not recognize that for x to be an integer, (10-k) must be divisible by 3, or they might incorrectly conclude what this means for the overall divisibility of x.

Errors while selecting the answer

1. Confusing "x must be a multiple of" with "x could be a multiple of"

Students might test whether x could sometimes be divisible by the given options rather than checking whether x is always divisible by them. For example, they might see that some values of x are divisible by 4 and incorrectly conclude that 4 is the answer.

2. Not checking all factor relationships of 20

After determining that \(\mathrm{x} = 20\mathrm{m}\), students might not systematically check which answer choices are factors of 20. They might miss that since \(20 = 2^2 \times 5\), x must be divisible by 10, but not necessarily by 3, 6, 7, or 8.

Alternate Solutions

Smart Numbers Approach

We can solve this systematically by finding valid values that satisfy our constraints and testing what x must be divisible by.

Step 1: Set up our constraint equation
We have: \(3\mathrm{x} + 4\mathrm{y} = 200\), where y is a multiple of 5

Step 2: Choose smart values for y
Since y must be a multiple of 5, let's test \(\mathrm{y} = 20\) (chosen because \(4 \times 20 = 80\), leaving 120 for \(3\mathrm{x}\), which gives us a clean integer for x).

Step 3: Calculate corresponding x
\(3\mathrm{x} + 4(20) = 200\)
\(3\mathrm{x} + 80 = 200\)
\(3\mathrm{x} = 120\)
\(\mathrm{x} = 40\)

Step 4: Test another smart value
Let's try \(\mathrm{y} = 35\):
\(3\mathrm{x} + 4(35) = 200\)
\(3\mathrm{x} + 140 = 200\)
\(3\mathrm{x} = 60\)
\(\mathrm{x} = 20\)

Step 5: Test one more value
Let's try \(\mathrm{y} = 5\):
\(3\mathrm{x} + 4(5) = 200\)
\(3\mathrm{x} + 20 = 200\)
\(3\mathrm{x} = 180\)
\(\mathrm{x} = 60\)

Step 6: Analyze the pattern
We found that x can be 40, 20, or 60. Let's check what these numbers are all divisible by:
• \(40 = 4 \times 10\)
• \(20 = 2 \times 10\)
• \(60 = 6 \times 10\)

Step 7: Identify the common factor
All our x values (40, 20, 60) are multiples of 10. Checking against our answer choices, we see that 10 is option E.

Step 8: Verify this makes sense
Since \(4\mathrm{y}\) is always a multiple of 20 (because y is a multiple of 5), and 200 is also a multiple of 20, then \(3\mathrm{x}\) must be a multiple of 20. This means x must be a multiple of \(\frac{20}{3}\). Since x must be an integer, and \(\gcd(3,20) = 1\), x must be a multiple of 20. However, our constraint is actually that x must be a multiple of 10, which we verified with our concrete examples.