TMUA Official 2020 Paper 2 · Question 3

A student makes the following claim:

For all integers $n$ , the expression $4\left(\dfrac{9n+1}2-\dfrac{3n-1}2\right)$ is divisible by 3.

Here is the student's argument:

\begin{aligned} 4\left(\frac{9n+1}2-\frac{3n-1}2\right)&=2\left(2\left(\frac{9n+1}2-\frac{3n-1}2\right)\right)&&\text{(I)}\\ &=2(9n+1-3n-1)&&\text{(II)}\\ &=2(6n)&&\text{(III)}\\ &=12n&&\text{(IV)}\\ &=3(4n)&&\text{(V)} \end{aligned}

which is always a multiple of 3. (VI)

So the expression $4\left(\dfrac{9n+1}2-\dfrac{3n-1}2\right)$ is always divisible by 3.

Which one of the following is true?

AThe argument is correct.

BThe argument is incorrect, and the first error occurs on line (I).

CThe argument is incorrect, and the first error occurs on line (II).

DThe argument is incorrect, and the first error occurs on line (III).

EThe argument is incorrect, and the first error occurs on line (IV).

FThe argument is incorrect, and the first error occurs on line (V).

GThe argument is incorrect, and the first error occurs on line (VI).

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