When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the...

1. Translate the problem requirements: Both x and y have identical remainder patterns (remainder 3 when divided by 5, remainder 4 when divided by 7). We need to find what must be a factor of their difference x - y.
2. Express the remainder conditions algebraically: Write both x and y in terms of their divisors and remainders to identify their mathematical relationship.
3. Recognize the identical pattern implication: Since both numbers have the same remainder conditions, determine what this tells us about their difference.
4. Apply the difference property: Use the fact that when two numbers have identical remainder patterns with respect to certain divisors, their difference has a predictable relationship to those divisors.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're told in everyday language. We have two positive integers, x and y, and they both follow the exact same pattern when we divide them by certain numbers.

For both x and y:

• When divided by 5, they leave a remainder of 3
• When divided by 7, they leave a remainder of 4

Think of this like two clocks that are set to show the same time - they might be different clocks, but they display identical information. Since \(\mathrm{x > y}\), we know x is the larger number, but both numbers follow identical remainder patterns.

The question asks: what must be a factor of \(\mathrm{x - y}\)? This means we need to find which of the answer choices will always divide evenly into the difference between any two numbers that follow this same remainder pattern.

Process Skill: TRANSLATE - Converting the remainder conditions into a clear mathematical relationship

2. Express the remainder conditions algebraically

Now let's write what we know using mathematical expressions. When we say "x divided by 5 gives remainder 3," we can write this as:

\(\mathrm{x = 5k + 3}\) (where k is some whole number)

Similarly, "x divided by 7 gives remainder 4" means:

\(\mathrm{x = 7m + 4}\) (where m is some whole number)

For y, since it has the exact same remainder pattern:

\(\mathrm{y = 5j + 3}\) (where j is some whole number)

\(\mathrm{y = 7n + 4}\) (where n is some whole number)

Notice that x and y have identical remainder parts (3 and 4), but different quotient parts (k vs j, and m vs n).

3. Recognize the identical pattern implication

Here's the key insight: since both x and y have identical remainder patterns, they belong to the same "family" of numbers.

Think of it this way - imagine all the positive integers that give remainder 3 when divided by 5 AND remainder 4 when divided by 7. These numbers form a sequence: some smallest number that works, then that number plus some fixed interval, then that number plus twice the interval, and so on.

Since x and y are both in this same sequence, the difference \(\mathrm{x - y}\) must be a multiple of the interval between consecutive terms in this sequence.

Process Skill: INFER - Recognizing that identical remainder patterns create a predictable difference structure

4. Apply the difference property

When two numbers have identical remainders with respect to certain divisors, their difference is divisible by those divisors.

Let's see this clearly:

• Since \(\mathrm{x = 5k + 3}\) and \(\mathrm{y = 5j + 3}\), then \(\mathrm{x - y = (5k + 3) - (5j + 3) = 5k - 5j = 5(k - j)}\)
• Since \(\mathrm{x = 7m + 4}\) and \(\mathrm{y = 7n + 4}\), then \(\mathrm{x - y = (7m + 4) - (7n + 4) = 7m - 7n = 7(m - n)}\)

This tells us:

• \(\mathrm{x - y}\) is divisible by 5
• \(\mathrm{x - y}\) is divisible by 7

Since \(\mathrm{x - y}\) is divisible by both 5 and 7, and these numbers share no common factors (5 and 7 are coprime), \(\mathrm{x - y}\) must be divisible by their product: \(\mathrm{5 × 7 = 35}\).

Looking at our answer choices, only 35 appears as option (E).

4. Final Answer

The answer is (E) 35.

We can verify this makes sense: any two numbers with identical remainder patterns when divided by 5 and 7 will have a difference that's always divisible by 35. For example, the smallest positive number satisfying our conditions is 18 (since \(\mathrm{18 = 5×3 + 3}\) and \(\mathrm{18 = 7×2 + 4}\)). The next such number is 53. Indeed, \(\mathrm{53 - 18 = 35}\), confirming our answer.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the relationship between x and y

Students often assume that since x and y have the same remainder conditions, they must be consecutive numbers in the sequence that satisfies these conditions. This leads them to try finding specific values for x and y rather than recognizing that \(\mathrm{x - y}\) could be any multiple of the common difference in this sequence.

2. Focusing on finding individual values instead of the difference pattern

Many students get sidetracked trying to solve for the exact values of x and y using the Chinese Remainder Theorem or by finding the least common solution. While this approach can work, it's unnecessarily complex and time-consuming. The key insight is that we only need to focus on what the difference \(\mathrm{x - y}\) must be divisible by, not the actual values.

3. Missing the connection between identical remainders and divisibility

Students may not immediately recognize that when two numbers have identical remainders when divided by the same divisors, their difference is automatically divisible by those divisors. This fundamental property is the cornerstone of the solution, and missing it leads to much more complicated approaches.

Errors while executing the approach

1. Incorrectly applying the LCM concept

When students recognize that \(\mathrm{x - y}\) must be divisible by both 5 and 7, they might incorrectly calculate the LCM. Since 5 and 7 are coprime (share no common factors), the LCM is simply \(\mathrm{5 × 7 = 35}\). However, students sometimes apply LCM formulas unnecessarily or make arithmetic errors in this step.

2. Algebraic manipulation errors

When working with the expressions \(\mathrm{x = 5k + 3}\), \(\mathrm{y = 5j + 3}\), etc., students may make errors in the subtraction step. For example, they might forget that \(\mathrm{(5k + 3) - (5j + 3) = 5k - 5j}\), incorrectly thinking the remainders don't cancel out, or they may make sign errors in the algebra.

Errors while selecting the answer

1. Selecting a factor of 35 instead of 35 itself

Students who correctly determine that \(\mathrm{x - y}\) must be divisible by 35 might then select answer choices like (A) 12 or (C) 20, thinking that if 35 divides \(\mathrm{x - y}\), then smaller numbers might also work. However, the question asks what MUST be a factor, and while 35 always divides \(\mathrm{x - y}\), its factors don't necessarily always divide \(\mathrm{x - y}\).

2. Confusing 'must be a factor' with 'could be a factor'

The question specifically asks which option 'must be a factor' of \(\mathrm{x - y}\). Some students might select answers that could work for specific values of x and y, but don't work for all possible pairs. For example, they might verify their answer with one example (like \(\mathrm{x = 53, y = 18}\)) but fail to check if their selected answer works universally for all valid x and y pairs.

Alternate Solutions

Smart Numbers Approach

Step 1: Find the smallest positive values for x and y

Since both x and y satisfy identical remainder conditions, we can find the smallest positive integer that satisfies these conditions and use it as our smart number.

For a number to have remainder 3 when divided by 5 and remainder 4 when divided by 7, we need to solve:

• \(\mathrm{n ≡ 3 \pmod{5}}\)
• \(\mathrm{n ≡ 4 \pmod{7}}\)

Testing values systematically:

• Numbers with remainder 3 when divided by 5: 3, 8, 13, 18, 23, 28, 33, 38...
• Check which also has remainder 4 when divided by 7:
  • \(\mathrm{3 ÷ 7 = 0 \text{ remainder } 3}\) ✗
  • \(\mathrm{8 ÷ 7 = 1 \text{ remainder } 1}\) ✗
  • \(\mathrm{13 ÷ 7 = 1 \text{ remainder } 6}\) ✗
  • \(\mathrm{18 ÷ 7 = 2 \text{ remainder } 4}\) ✓

So 18 is the smallest positive integer satisfying both conditions.

Step 2: Choose smart values for x and y

Since both x and y have identical remainder patterns, they must both be of the form \(\mathrm{18 + 35k}\) for some integer \(\mathrm{k ≥ 0}\).

Let's choose:

• \(\mathrm{y = 18}\) (the smallest possible value)
• \(\mathrm{x = 18 + 35 = 53}\) (the next possible value)

Step 3: Verify our choices satisfy all conditions

For \(\mathrm{y = 18}\):

• \(\mathrm{18 ÷ 5 = 3 \text{ remainder } 3}\) ✓
• \(\mathrm{18 ÷ 7 = 2 \text{ remainder } 4}\) ✓

For \(\mathrm{x = 53}\):

• \(\mathrm{53 ÷ 5 = 10 \text{ remainder } 3}\) ✓
• \(\mathrm{53 ÷ 7 = 7 \text{ remainder } 4}\) ✓

Step 4: Calculate x - y and check factors

\(\mathrm{x - y = 53 - 18 = 35}\)

Now check which answer choices are factors of 35:

• (A) 12: \(\mathrm{35 ÷ 12 = 2.92...}\) ✗
• (B) 15: \(\mathrm{35 ÷ 15 = 2.33...}\) ✗
• (C) 20: \(\mathrm{35 ÷ 20 = 1.75}\) ✗
• (D) 28: \(\mathrm{35 ÷ 28 = 1.25}\) ✗
• (E) 35: \(\mathrm{35 ÷ 35 = 1}\) ✓

Step 5: Verify with another pair

To ensure this isn't coincidental, let's try another pair:

• \(\mathrm{y = 53, x = 53 + 35 = 88}\)
• \(\mathrm{x - y = 88 - 53 = 35}\)

The difference is always 35, confirming that 35 must be a factor of \(\mathrm{x - y}\).

Answer: (E) 35