When the even integer n is divided by 9, the remainder is 8. Which of the following, when added to...

1. Translate the problem requirements: We need to understand what "\(\mathrm{n}\) divided by 9 has remainder 8" means, recognize that \(\mathrm{n}\) is even, and find which number when added to \(\mathrm{n}\) creates a sum divisible by 18 (remainder 0 when divided by 18).
2. Express n in terms of its remainder relationship: Write \(\mathrm{n}\) in the form that captures the division by 9 with remainder 8, ensuring \(\mathrm{n}\) remains even.
3. Set up the divisibility condition: Determine what \((\mathrm{n} + \text{added number})\) must equal for the sum to be divisible by 18.
4. Test each answer choice systematically: Check which added number makes \((\mathrm{n} + \text{that number})\) divisible by 18 using the remainder pattern.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're told in everyday language:

• "\(\mathrm{n}\) is an even integer" - This means \(\mathrm{n}\) could be numbers like 2, 4, 6, 8, 10, 12, etc.
• "When \(\mathrm{n}\) is divided by 9, the remainder is 8" - This means if we divide \(\mathrm{n}\) by 9, we get some whole number plus 8 left over. For example, if \(\mathrm{n} = 17\), then \(17 ÷ 9 = 1\) remainder 8.
• "Which number, when added to \(\mathrm{n}\), gives a sum divisible by 18?" - We need to find what number to add to \(\mathrm{n}\) so that the total divides evenly by 18 (no remainder).

But wait - there's something important to notice here. If \(\mathrm{n}\) divided by 9 gives remainder 8, then \(\mathrm{n}\) could be 8, 17, 26, 35, etc. But \(\mathrm{n}\) must be EVEN, so \(\mathrm{n}\) can only be 8, 26, 44, 62, etc.

2. Express n in terms of its remainder relationship

Since \(\mathrm{n}\) leaves remainder 8 when divided by 9, we can write \(\mathrm{n}\) in a helpful way:

\(\mathrm{n} = 9\mathrm{k} + 8\), where \(\mathrm{k}\) is some whole number

But remember, \(\mathrm{n}\) must be even. Let's check: if \(\mathrm{n} = 9\mathrm{k} + 8\), when is this even?

• If \(\mathrm{k} = 0\): \(\mathrm{n} = 8\) (even) ✓
• If \(\mathrm{k} = 1\): \(\mathrm{n} = 17\) (odd) ✗
• If \(\mathrm{k} = 2\): \(\mathrm{n} = 26\) (even) ✓
• If \(\mathrm{k} = 3\): \(\mathrm{n} = 35\) (odd) ✗

The pattern shows \(\mathrm{n}\) is even when \(\mathrm{k}\) is even. So we can write: \(\mathrm{n} = 9(2\mathrm{m}) + 8 = 18\mathrm{m} + 8\) for some whole number \(\mathrm{m}\).

This means \(\mathrm{n} = 8, 26, 44, 62, 80\), etc.

3. Set up the divisibility condition

We want \((\mathrm{n} + \text{added number})\) to be divisible by 18.

Since \(\mathrm{n} = 18\mathrm{m} + 8\), we have:

\(\mathrm{n} + \text{added number} = (18\mathrm{m} + 8) + \text{added number} = 18\mathrm{m} + (8 + \text{added number})\)

For this to be divisible by 18, the part \((8 + \text{added number})\) must be divisible by 18.

In plain English: since \(\mathrm{n}\) is already "\(18\mathrm{m} + 8\)", we need to add something to that 8 to make it reach the next multiple of 18.

The multiples of 18 are: 0, 18, 36, 54...

Since we have 8, we need \(8 + \text{added number} = 18\) (the next multiple)

So: \(\text{added number} = 18 - 8 = 10\)

4. Test each answer choice systematically

Let's verify by testing with a concrete example. Let's use \(\mathrm{n} = 26\) (which is \(18×1 + 8\)):

• Choice A (1): \(26 + 1 = 27\). Is \(27 ÷ 18 = 1\) remainder 9? Yes, so not divisible by 18. ✗
• Choice B (4): \(26 + 4 = 30\). Is \(30 ÷ 18 = 1\) remainder 12? Yes, so not divisible by 18. ✗
• Choice C (9): \(26 + 9 = 35\). Is \(35 ÷ 18 = 1\) remainder 17? Yes, so not divisible by 18. ✗
• Choice D (10): \(26 + 10 = 36\). Is \(36 ÷ 18 = 2\) remainder 0? Yes! This IS divisible by 18. ✓
• Choice E (17): \(26 + 17 = 43\). Is \(43 ÷ 18 = 2\) remainder 7? Yes, so not divisible by 18. ✗

Let's double-check with another value, \(\mathrm{n} = 8\):

\(8 + 10 = 18\), and \(18 ÷ 18 = 1\) remainder 0 ✓

4. Final Answer

The answer is D. 10

When we add 10 to any even integer \(\mathrm{n}\) that has remainder 8 when divided by 9, the sum will always be divisible by 18. This works because such numbers have the form \(18\mathrm{m} + 8\), and adding 10 gives us \(18\mathrm{m} + 18 = 18(\mathrm{m} + 1)\), which is clearly divisible by 18.

Common Faltering Points

Errors while devising the approach

1. Missing the "even integer" constraint: Students often focus only on the remainder condition (\(\mathrm{n} ÷ 9 =\) remainder 8) and forget that \(\mathrm{n}\) must be even. This leads them to consider all numbers of the form \(9\mathrm{k} + 8\) (like 8, 17, 26, 35...) instead of only the even ones (8, 26, 44, 62...). This constraint is crucial because it changes the pattern from \(\mathrm{n} = 9\mathrm{k} + 8\) to \(\mathrm{n} = 18\mathrm{m} + 8\).

2. Misunderstanding what "divisible by 18" means: Some students confuse "divisible by 18" with "divisible by 9" or think it means the sum should have remainder 18 when divided by some number. They need to clearly understand that divisible by 18 means the remainder is 0 when divided by 18.

3. Not recognizing the modular arithmetic pattern: Students may try to solve this by testing random values instead of recognizing that if \(\mathrm{n} = 18\mathrm{m} + 8\), then for \((\mathrm{n} + \mathrm{x})\) to be divisible by 18, we need \((8 + \mathrm{x})\) to be divisible by 18, which gives us \(\mathrm{x} = 10\).

Errors while executing the approach

1. Arithmetic errors when checking the even constraint: When determining which values of \(\mathrm{k}\) make \(9\mathrm{k} + 8\) even, students might incorrectly conclude that \(\mathrm{k}\) must be odd instead of even, or make calculation errors when checking specific values like \(\mathrm{n} = 17, 26, 35\).

2. Incorrect substitution in the general form: Students may correctly identify that \(\mathrm{n} = 18\mathrm{m} + 8\) but then make algebraic errors when setting up the divisibility condition, such as writing \((\mathrm{n} + \mathrm{x}) = 18\mathrm{m} + 8 + \mathrm{x}\) incorrectly or forgetting to group terms properly.

Errors while selecting the answer

1. Verification errors with test values: Students might correctly identify that the answer should be 10, but then make arithmetic mistakes when checking their answer with specific values of \(\mathrm{n}\) (like calculating \(26 + 10 = 35\) instead of 36, or incorrectly determining that 36 is not divisible by 18).

2. Selecting the first option that "seems to work": Without systematic testing, students might test answer choices in order and select an incorrect option if they make a calculation error, rather than verifying all choices or double-checking their work with multiple test values.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a smart number for n

Since \(\mathrm{n}\) is even and leaves remainder 8 when divided by 9, let's find the smallest such number:

• When we divide by 9, remainder is 8, so \(\mathrm{n} = 9\mathrm{k} + 8\) for some integer \(\mathrm{k}\)
• For \(\mathrm{n}\) to be even, we need \(9\mathrm{k} + 8\) to be even
• Since 8 is even, we need \(9\mathrm{k}\) to be even, which means \(\mathrm{k}\) must be even
• Let \(\mathrm{k} = 0\): \(\mathrm{n} = 9(0) + 8 = 8\)
• Check: \(8 ÷ 9 = 0\) remainder 8 ✓, and 8 is even ✓

Step 2: Test each answer choice with n = 8

We need \((\mathrm{n} + \text{added number})\) to be divisible by 18.

Choice A: \(8 + 1 = 9\)

\(9 ÷ 18 = 0\) remainder 9. Not divisible by 18.

Choice B: \(8 + 4 = 12\)

\(12 ÷ 18 = 0\) remainder 12. Not divisible by 18.

Choice C: \(8 + 9 = 17\)

\(17 ÷ 18 = 0\) remainder 17. Not divisible by 18.

Choice D: \(8 + 10 = 18\)

\(18 ÷ 18 = 1\) remainder 0. Divisible by 18! ✓

Choice E: \(8 + 17 = 25\)

\(25 ÷ 18 = 1\) remainder 7. Not divisible by 18.

Step 3: Verify with another smart number

Let's try \(\mathrm{k} = 2\): \(\mathrm{n} = 9(2) + 8 = 26\)

Check: 26 is even ✓, and \(26 ÷ 9 = 2\) remainder 8 ✓

Testing choice D: \(26 + 10 = 36\)

\(36 ÷ 18 = 2\) remainder 0. Divisible by 18! ✓

Answer: D