TMUA 2020 · Paper 2 · Question 3

A student makes the following claim:

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

Here is the student's argument:

4 (\frac{9 n + 1}{2} - \frac{3 n - 1}{2}) = 2 (2 (\frac{9 n + 1}{2} - \frac{3 n - 1}{2})) (I)

= 2 (9 n + 1 - 3 n - 1) (II)

= 2 (6 n) (III)

= 12 n (IV)

= 3 (4 n) (V)

which is always a multiple of 3. \quad \text{(VI)}

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

Which one of the following is true?