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Proof, EXT2 P1 2024 HSC 14a

Prove that if \(a\) is any odd integer, then  \(a^2-1\)  is divisible by 8.   (2 marks) 

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\(\text {Prove if  \(a\) is odd, \(a^2-1\) is divisible by 8.}\)

\(\text {Let}\ \ a=2 n+1, \ \ n \in \mathbb{Z}\)

  \(a^2-1\) \(=(2 n+1)^2-1\)
    \(=4 n^2+4 n+1-1\)
    \(=4 n(n+1)\)

 
\(\text{If \(n\) is odd (\(n=2 k+1, k \in Z)\):}\)

\(a^2-1=4(2 k+1)(2 k+2)=8(2 k+1)(k+1) /8\)

\(\text{If \(n\) is even \((n=2 k)\):}\)

\(a^2-1=4(2 k)(2 k+1)=8 k(2 k+1) /8\)

\(\therefore \text{ If \(a\) is odd, \(a^2-1\) is divisible by 8}\)

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\(\text {Prove if  \(a\) is odd, \(a^2-1\) is divisible by 8.}\)

\(\text {Let}\ \ a=2 n+1, \ \ n \in \mathbb{Z}\)

  \(a^2-1\) \(=(2 n+1)^2-1\)
    \(=4 n^2+4 n+1-1\)
    \(=4 n(n+1)\)

 
\(\text{If \(n\) is odd (\(n=2 k+1, k \in Z)\):}\)

\(a^2-1=4(2 k+1)(2 k+2)=8(2 k+1)(k+1) /8\)

\(\text{If \(n\) is even \((n=2 k)\):}\)

\(a^2-1=4(2 k)(2 k+1)=8 k(2 k+1) /8\)

\(\therefore \text{ If \(a\) is odd, \(a^2-1\) is divisible by 8}\)

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 2021 HSC 12b

Consider Statement A.

Statement A: ‘If `n^2` is even, then `n` is even.’

  1. What is the converse of Statement A?.  (1 mark)

  2. Show that the converse of Statement A is true. (1 mark)

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  1. `text{See Worked Solution}`
  2. `text{See Worked Solution}`
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i.     `text{Converse}`

`text{If} \ n \ text{is even, then} \ n^2 \ text{is even.}`
 

ii.    `text{If} \ n \ text{is even:}`

`n` `= 2p, \ p  ∈  ZZ`
`n^2` `= (2 p)^2`
  `=4 p^2`
  `= 2 (2p^2)`
  `=2q, \ q  ∈  ZZ`

 

`:. \ text{If} \ n \ text{is even, then} \ n^2 \ text{is even.}`

Proof, EXT2 P1 SM-Bank 13

If  `(n - 3)^2`  is an even integer, prove by contrapositive that  `n`  is odd.   (2 marks)

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`text{Proof (See Worked Solutions)}`

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`text(Statement)`

`text(If) \ \ (n – 3)^2 \ \ text(is even) => n \ text(is odd)`

`text(Contrapositive)`

`text(If) \ \ n \ not \ text(odd) => (n – 3)^2 \ not \ text(even)`

`text{(i.e.)}\ \  \ n \ text(even) => (n – 3)^2 \ text(is odd)`
 

`text(If)\ \ n \ \ text(even), \ ∃ \ k, \ k ∈ Ζ \ \ text(where)\ \ n = 2k`

`(n – 3)^2` `= (2k – 3)^2`
  `= 4k^2 – 12k + 9`
  `= 4(k^2 – 12k + 2) + 1`

 
`=> (n – 3)^2 \ \ text(is odd)`
 
`:. \ text(If) \ \ n\  \ text(is even), (n – 3)^2 \ \ text(is odd)`

`:. \ text(By contrapositive, statement is true.)`

Proof, EXT2 P1 SM-Bank 11

If  `a^2-4a + 3`  is even, `a ∈ Ζ`,

prove by contrapositive that  `a`  is odd. (3 marks)

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`text{Proof (See Worked Solutions)}`

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`text(Proof by contrapositive)`

`a not \ text(odd)` `\ =>\ \ a^2-4x + 3 not \ text(even)`
`text{(i.e)  If} \ \ a \ text(is even) ` `\ => \ a^2-4x + 3 \ \ text(is odd)`

 

`text(If) \ \ a \ \ text(is even) , ∃ \ k , k ∈ Ζ , text(such that) \ \ a = 2k`

`text(Substitute) \ \ 2k \ \ text(into) \ \ a^2-4x + 3`

`(2k)^2-4(2k) + 3` `= 4k^2-8k + 3`
  `= 2(2k^2-4k + 1) + 1`

 
`:. \ text(By contrapositive, if) \ \ a^2-4x + 3 \ \ text(is even) \ =>  \ a \ \ text(is odd.)`