EXM.1.SL.TZ0.1

pestleMathematicsAASLPaper 1EXM· sl-1-6-simple-proofsource ↗

Explain why any integer can be written in the form $4k$ or $4k + 1$ or $4k + 2$ or $4k + 3$ , where $k \in \mathbb{Z}$ .

[2]

Hence prove that the square of any integer can be written in the form $4t$ or $4t + 1$ , where $t \in {\mathbb{Z}^ + }$ .

[6]

Markscheme / solution

Upon division by 4 M1

any integer leaves a remainder of 0, 1, 2 or 3. R1

Hence, any integer can be written in the form $4k$ or $4k + 1$ or $4k + 2$ or $4k + 3$ , where $k \in \mathbb{Z}$ AG

[2 marks]

${\left( {4k} \right)^2} = 16{k^2} = 4t$ M1A1

${\left( {4k + 1} \right)^2} = 16{k^2} + 8k + 1 = 4t + 1$ M1A1

${\left( {4k + 2} \right)^2} = 16{k^2} + 16k + 4 = 4t$ A1

${\left( {4k + 3} \right)^2} = 16{k^2} + 24k + 9 = 4t + 1$ A1

Hence, the square of any integer can be written in the form $4t$ or $4t + 1$ , where $t \in {\mathbb{Z}^ + }$ . AG

[6 marks]

Examiners’ report

[N/A]