本文发表在 rolia.net 枫下论坛Technical Score: 6 / 7
Style Score: 0.8 / 1
Comments:
Part (a)
Good job! You showed that the expression can be written as a sum of three multiples of $6$, which implied that the expression must also be divisible by $6$.
You proved that the product of three consecutive numbers must be divisible by $2$ and that it must be divisible by $3$ for all $n.$ This is a great way to show that it must be divisible by $2 \cdot 3 = 6$ for all $n$. However, it might be good to clearly state that this can be done since $2$ and $3$ are coprime.
Although your solution is good, there was a much easier way to prove this. For example, using the same kind of reasoning, you could have proven that the initial expression is divisible by $2$ and $3$, which means that it is divisible by $6$. Can think of a way to prove this without involving products of consecutive numbers? Hint.
You can take different cases, taking into account the remainder you can get when you divide $n$ by $3$.
Part (b)
You got the correct answer and it looks like you had some good ideas for this part. You were on the right track by thinking that in order for the expression to be divisible by $12,$ it has to be divisible by $3$ and $4$. However, some parts of your solution were a bit confusing. Let's have a look.
Note that if one of the factors is a multiple of $3$, the product of the other factors does not necessarily have to be a multiple of $4$. For example, you could have $n=12$. Be careful with the terminology you use in your solution, to avoid this kind of confusion. Also, the next parts are not very well explained and it is not quite clear what you meant by those. Make sure your solution is well explained, so that the reader can see how and why you go from one step to another.
Think about this part a bit more. Can you explain why the expression is always divisible by $3?$ Think about the result you got in part (a). What else do you need to check to make sure the expression is divisible by $4?$ Try to take different cases, using the remainder you get when $n$ is divided by $4$.
General comments
In terms of style, your solution is generally easy to read and looks organized, since you used short, clear sentences and you have separated different ideas into separate paragraphs. Good job! However, in the end of part (b), your solution is quite hard to follow. Make sure you explain every step of your proof in detail.
Good job using "we" statements to make your solution sound more professional! However, avoid using other pronouns such as "you".
The use of $\LaTeX$ makes your work look professional, equations look neat and the layout is organized. Keep it up! You also placed important equations and expressions on their own separate lines making your work easy to read. Excellent! However, note that if you write an equation between double dollar signs it appears centered on a separate line, making it much easier to read.
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