Q15 (a) (iii)

Criteria	Marks
Provides correct solution	3
Considers cases to justify their statement, or equivalent merit	2
Correctly states the converse or equivalent merit	1

Sample answer:

The converse states that:

If $k^3+1$ is divisible by 3, then $(k+1)$ is divisible by 3.

This statement is true.

The integer $k$ must be of the form $3j$, $3j+1$ or $3j-1$, for some integer $j$.

Then $k^3+1=27j^3+1$ or $(3j+1)^3+1$ or $(3j-1)^3+1$.

Only the third case gives $k^3+1$ divisible by 3.

OR

Using proof by contradiction, suppose $k^3+1$ is divisible by 3.

But $(k+1)$ is not divisible by 3.

$$\therefore k^2-k+1\text{ is divisible by 3}$$ $$\therefore(k+1)^2-3k\text{ is divisible by 3}$$ $$\therefore(k+1)^2\text{ and hence }(k+1)\text{ is divisible by 3}.$$

This is a contradiction.