smc-7422-55-Primes

Proof, EXT2 P1 2022 HSC 13a

Prove that for all integers `n` with `n >= 3`, if `2^(n)-1` is prime, then `n` cannot be even. (3 marks)

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`text{Proof (See Worked Solutions)}`

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`text{Contrapositive statement:}`

`text{If}\ n\ text{is even}, 2^n-1\ text{is NOT prime.}`

`text{Let}\ \ n=2k,\ \ (kinZZ and k>=2)`

`2^n-1`	`=2^(2k)-1`
	`=(2^k)^2-1`
	`=(2^k-1)(2^k+1)`

`text{S}text{ince}\ \ k>=2\ \ =>\ \ 2^k-1>=3 and 2^k+1>=5`

`:.2^n-1\ \ text{is not prime if}\ n\ text{is even, as it has two non-trivial integer factors.}`

`:.\ text{By contrapositive statement, if}\ 2^(n)-1\ text{is prime}, n\ text{cannot be even.}`

Proof, EXT2 P1 2022 HSC 7 MC

Consider the statement `P`.

`P` : For all integers `n \geq 1`, if `n` is a prime number then `(n(n+1))/(2)` is a prime number.

Which of the following is true about this statement and its converse?

The statement `P` and its converse are both true.
The statement `P` and its converse are both false.
The statement `P` is true and its converse is false.
The statement `P` is false and its converse is true.

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`D`

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`text{Statement:}\ \ ∀n in ZZ^+, text{if}\ n\ text{is prime}\ \ =>\ \ (n(n+1))/(2)\ text{is prime.}`

`text{Converse:}\ \ ∀n in ZZ^+, text{if}\ (n(n+1))/(2)\ text{is prime}\ \ =>\ \ n\ text{is prime.}`

`text{Consider prime}\ n=3:`

`(n(n+1))/(2)=(3xx4)/2=6\ \ text{(not prime → Statement is false)}`

`text{Consider}\ \ (n(n+1))/(2):`

`(n(n+1))/(2)\ text{is a prime}\ iff\ n=2\ \ text{(prime)}`

`:.\ text{Converse is true}`

`=>D`

♦♦ Mean mark 37%.

Proof, EXT2 P1 2020 HSC 7 MC

Consider the proposition:

'If `2^n-1` is not prime, then `n` is not prime'.

Given that each of the following statements is true, which statement disproves the proposition?

`2^5-1` is prime
`2^6-1` is divisible by 9
`2^7-1` is prime
`2^11-1` is divisible by 23

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`D`

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`text(Strategy 1 – Contradiction)`

`text(Consider option)\ D,`

`text(S)text(ince)\ \ 2^11 -1\ \ text(is divisible by 23, it is NOT prime.)`

`text(The proposition states that 11 is not prime which is false.)`

`:. 2^11-1\ \ text(is divisible by 23, disproves the proposition.)`

`text(Strategy 2 – Contrapositive)`

`text{The proposition is conditional}`

`X => Y`

`text{L}text{ogically equivalent contrapositive statement}`

`not \ Y => not \ X`

`text{i.e. If} \ n \ text{is prime} \ => \ 2^n-1 \ text{is prime.}`

`text{Consider D:}`

`n = 11 \ text{(prime)}`

`2^11-1 \ text{is divisible by 23 (not prime)}`

`therefore \ text{Contrapositive statement is false and disproves the proposition.}`

`=> \ D`