Prove that if $a$ and $b$ are even integers, then $a + b$ is even.
Question HKC78-AASL-PRF-01
[Maximum mark: 4]
Prove that if $a$ and $b$ are even integers, then $a + b$ is even.
Mark Scheme
Let the even integers be $2a$ and $2b$, where $a$ and $b$ are integers. [1]
Their sum is $2a + 2b = 2(a + b)$. [2]
$a + b$ is an integer, so the sum is even. [1]
Total: 4 marks
Prove that the square of an even integer is divisible by $4$.
Question HKC78-AASL-PRF-02
[Maximum mark: 4]
Prove that the square of an even integer is divisible by $4$.
Mark Scheme
Let the even integer be $n = 2k$, where $k$ is an integer. [1]
$n^{2} = 4k^{2}$. [2]
$k^{2}$ is an integer, so $4$ divides $n^{2}$. [1]
Total: 4 marks
