In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^3 + 1 is divisible by 3'. (i) Prove that the proposition P is true. (ii) Write down the contrapositive of the proposition P. (iii) Write down the converse of the proposition P and state, with reasons, whether this converse is true or false.

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In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^3 + 1 is divisible by 3' - HSC - SSCE Mathematics Extension 2 - Question 15 - 2020 - Paper 1

Question 15

In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^3 + 1 is divisible by 3'. (i) Prove that the proposition P is true. (ii) Wri... show full transcript

Worked Solution & Example Answer:In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^3 + 1 is divisible by 3' - HSC - SSCE Mathematics Extension 2 - Question 15 - 2020 - Paper 1

Step 1

(i) Prove that the proposition P is true.

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Answer

To prove the proposition, we start by assuming that $k + 1$ is divisible by 3. This means there exists some integer $j$ such that: $k + 1 = 3j$ Now, we need to determine if $k^3 + 1$ is also divisible by 3:

Substitute $k = 3j - 1$ to obtain: $k^3 + 1 = (3j - 1)^3 + 1$
Expanding this using the binomial theorem gives: $(3j - 1)^3 = 27j^3 - 27j^2 + 9j - 1$ Thus: $k^3 + 1 = 27j^3 - 27j^2 + 9j$
The expression can be factored to show: $k^3 + 1 = 3(9j^3 - 9j^2 + 3j)$

Since the result is a multiple of 3, we conclude that $k^3 + 1$ is divisible by 3, hence the proposition $P$ is true.

Step 2

(ii) Write down the contrapositive of the proposition P.

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Answer

The contrapositive of the proposition P is:

'If $k^3 + 1$ is not divisible by 3, then $k + 1$ is not divisible by 3.'

Step 3

(iii) Write down the converse of the proposition P and state, with reasons, whether this converse is true or false.

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Answer

The converse of the proposition P is:

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

To determine whether this is true, assume that $k^3 + 1$ is divisible by 3. If we let $k^3 + 1 = 3m$ for some integer $m$ , this does not necessarily imply that $k + 1$ must also be divisible by 3.

For instance, consider $k = 2$ . Here:

$k^3 + 1 = 2^3 + 1 = 8 + 1 = 9$ , which is divisible by 3.
However, $k + 1 = 2 + 1 = 3$ , which is also divisible by 3.

Now consider $k = 4$ :

$k^3 + 1 = 4^3 + 1 = 64 + 1 = 65$ , which is not divisible by 3.
$k + 1 = 4 + 1 = 5$ , which is also not divisible by 3.

However, if we try $k = 1$ :

$k^3 + 1 = 1^3 + 1 = 1 + 1 = 2$ , not divisible by 3.
But $k + 1 = 2$ , still not consistent.

Thus, various values satisfy or contradict the converse, leading to the conclusion: the converse is not universally true.

In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^3 + 1 is divisible by 3' - HSC - SSCE Mathematics Extension 2 - Question 15 - 2020 - Paper 1

Worked Solution & Example Answer:In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^3 + 1 is divisible by 3' - HSC - SSCE Mathematics Extension 2 - Question 15 - 2020 - Paper 1

(i) Prove that the proposition P is true.

(ii) Write down the contrapositive of the proposition P.

(iii) Write down the converse of the proposition P and state, with reasons, whether this converse is true or false.

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