If d represents a digit from 0 through 9, what is the remainder when the number d,340 is divided by...

1. Translate the problem requirements: We need to find the remainder when \(\mathrm{d,340}\) (a 4-digit number where d is any digit 0-9) is divided by 8. The comma represents a thousands separator, so this is the number \(\mathrm{1000d + 340}\).
2. Apply the divisibility rule for 8: Since 1000 is divisible by 8 (\(\mathrm{1000 ÷ 8 = 125}\)), the remainder depends only on the last three digits of any number when divided by 8.
3. Focus on the constant portion: Since \(\mathrm{1000d}\) contributes no remainder, we only need to find the remainder when 340 is divided by 8, regardless of the value of d.
4. Calculate the final remainder: Perform the division \(\mathrm{340 ÷ 8}\) to determine the remainder.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find. The problem gives us a number written as "\(\mathrm{d,340}\)" where d can be any digit from 0 through 9. The comma here is just a thousands separator (like when we write 1,000 or 5,340), so this represents a four-digit number.

For example:

• If d = 1, the number is 1,340
• If d = 5, the number is 5,340
• If d = 0, the number is 0,340 (which is just 340)

Mathematically, this number can be written as: \(\mathrm{1000d + 340}\), where d is our unknown digit.

We need to find the remainder when this number is divided by 8, regardless of what digit d represents.

Process Skill: TRANSLATE - Converting the comma notation into mathematical understanding

2. Apply the divisibility rule for 8

Here's a very useful property about division by 8: when you divide any number by 8, only the last three digits of that number affect the remainder.

Why does this work? Because 1000 divides evenly by 8:

\(\mathrm{1000 ÷ 8 = 125}\) exactly (no remainder)

This means that 1000, 2000, 3000, 4000, etc. all leave remainder 0 when divided by 8.

So when we have our number \(\mathrm{1000d + 340}\):

• The part \(\mathrm{1000d}\) will always give remainder 0 when divided by 8 (since \(\mathrm{1000d}\) is a multiple of 1000)
• Only the 340 part will contribute to the final remainder

3. Focus on the constant portion

Since we've established that \(\mathrm{1000d}\) contributes no remainder regardless of what d is, we can ignore the thousands digit completely!

This means:

• Whether our number is 1,340 or 5,340 or 9,340
• They will ALL have the same remainder when divided by 8
• That remainder is the same as dividing just 340 by 8

This is why the answer won't be "Cannot be determined" - even though d can vary, the remainder stays constant.

4. Calculate the final remainder

Now we simply need to find the remainder when 340 is divided by 8.

Let's do this division step by step:

\(\mathrm{340 ÷ 8 = ?}\)

We can think: what's the largest multiple of 8 that fits into 340?

\(\mathrm{8 × 40 = 320}\)

\(\mathrm{8 × 42 = 336}\)

\(\mathrm{8 × 43 = 344}\) (this is too big)

So \(\mathrm{8 × 42 = 336}\) is the largest multiple of 8 that doesn't exceed 340.

The remainder is: \(\mathrm{340 - 336 = 4}\)

We can verify: \(\mathrm{340 = 8 × 42 + 4}\)

5. Final Answer

The remainder when \(\mathrm{d,340}\) is divided by 8 is 4, regardless of the value of digit d.

The answer is B. 4

Common Faltering Points

Errors while devising the approach

Students may incorrectly interpret "\(\mathrm{d,340}\)" as a decimal number (like d.340) rather than understanding it as a thousands separator representing a four-digit number. This fundamental misunderstanding would lead them down the wrong path entirely, possibly thinking they need to work with fractional values.

Seeing that d can be any digit from 0-9, students might immediately conclude that without knowing the specific value of d, the remainder cannot be determined. They fail to recognize that mathematical properties (like divisibility rules) can sometimes make variable portions irrelevant to the final answer.

Instead of recognizing the underlying mathematical principle about divisibility by 8, students might plan to calculate the remainder for all 10 possible values of d (from 0,340 to 9,340). While this approach would eventually work, it's inefficient and misses the elegant mathematical insight.

Errors while executing the approach

Students might know that divisibility by 8 depends on the last three digits but incorrectly think this means "the last three digits must be divisible by 8 for the whole number to be divisible by 8." They fail to understand that this rule applies to finding remainders, not just checking divisibility.

When calculating \(\mathrm{340 ÷ 8}\), students might make computational mistakes. For example, they might incorrectly calculate \(\mathrm{8 × 42 = 326}\) instead of 336, leading to a wrong remainder of 14 instead of 4. Alternatively, they might correctly get \(\mathrm{8 × 42 = 336}\) but then calculate \(\mathrm{340 - 336 = 6}\) instead of 4.

Errors while selecting the answer

After correctly calculating that \(\mathrm{340 ÷ 8 = 42}\) remainder 4, students might mistakenly select an answer choice that corresponds to the quotient (42) rather than the remainder (4). Since 42 isn't among the choices, they might look for a related number or become confused about what the question is asking.