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If \(\mathrm{n}\) is a negative integer and \(\mathrm{q}\) is a positive integer, then there is only one negative integer \(\mathrm{m}\) such that \(\mathrm{n = mq + r}\), where \(\mathrm{0 ≤ r ≤ q}\). The Integer \(\mathrm{r}\) is called the remainder when \(\mathrm{n}\) is divided by \(\mathrm{q}\). What is remainder when \(\mathrm{-100}\) is divided by \(\mathrm{30}\)?
Let's start by understanding what this problem is really asking in plain English. We need to find a way to write -100 as:
\(-100 = \text{(some negative integer)} \times 30 + \text{(some remainder)}\)
The catch is that the remainder must be between 0 and 30 (including 0, but not exceeding 30). This is different from how we normally think about division with negative numbers.
In everyday division, we might say \(-100 \div 30 = -3.33...\), but here we need to be more careful. We need to find a negative integer m and a non-negative remainder r such that:
\(-100 = \mathrm{m} \times 30 + \mathrm{r}\), where \(0 \leq \mathrm{r} \leq 30\)
Process Skill: TRANSLATE - Converting the formal mathematical language into a clear requirement
Let's think about this step by step. If we were dividing 100 by 30 (ignoring the negative sign for a moment), we'd get:
\(100 = 3 \times 30 + 10\)
This tells us that \(100 = 90 + 10\), which checks out.
But our number is -100, not 100. If we try to use the same approach:
\(-100 = -3 \times 30 + (-10)\)
This gives us \(-100 = -90 + (-10) = -100\) ✓
However, there's a problem! Our remainder is -10, but the problem requires the remainder to be non-negative (between 0 and 30).
Since our remainder of -10 doesn't satisfy the constraint \(0 \leq \mathrm{r} \leq 30\), we need to adjust our approach.
Here's the key insight: we can "borrow" from the quotient to make the remainder positive. Think of it like this:
Instead of: \(-100 = -3 \times 30 + (-10)\)
We can write: \(-100 = -4 \times 30 + 20\)
Let's check this: \(-4 \times 30 + 20 = -120 + 20 = -100\) ✓
Now our remainder is 20, which satisfies our constraint because \(0 \leq 20 \leq 30\).
This works because we're essentially saying: "Instead of being 3 groups of 30 short (with 10 extra shortage), we're 4 groups of 30 short (with 20 left over)."
Process Skill: APPLY CONSTRAINTS - Ensuring our solution meets all given requirements
Let's double-check our answer:
Therefore, the remainder when -100 is divided by 30 is 20.
The remainder when -100 is divided by 30 is 20.
This corresponds to answer choice E.
Students often miss that the remainder r must satisfy \(0 \leq \mathrm{r} \leq \mathrm{q}\), meaning it must be non-negative. They might think they can use standard division rules where remainders can be negative, leading them to accept r = -10 as a valid remainder.
The problem specifically states that m must be a negative integer, but students might overlook this constraint. They could attempt to use a positive quotient, which would violate the given conditions.
Students might not recognize that this is asking for the division algorithm format \(\mathrm{n} = \mathrm{mq} + \mathrm{r}\) and instead try to use regular decimal division \((-100 \div 30 = -3.33...)\), missing the requirement for integer quotient and specific remainder constraints.
After finding \(-100 = -3 \times 30 + (-10)\), students might accept r = -10 as the final answer without checking if it satisfies the non-negative remainder constraint \((0 \leq \mathrm{r} \leq 30)\).
When students realize they need to make the remainder positive, they might incorrectly calculate the adjustment. For example, they might try \(-100 = -2 \times 30 + (-40)\) instead of properly "borrowing" one more group of 30 to get \(-100 = -4 \times 30 + 20\).
Students might make calculation mistakes when checking their work, such as computing \((-4) \times 30 + 20\) incorrectly, leading them to doubt their correct answer or accept an incorrect one.
Even after working through the problem, students might select answer choice B (-10) because they remember getting -10 during their initial calculation and don't recognize that this doesn't satisfy the remainder constraints.
Students might accidentally select the quotient value (-4) if it were among the choices, or get confused about which value represents the remainder r versus the quotient m in their final equation.