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Date May Example questions Marks available 2 Reference code EXM.1.SL.TZ0.1
Level Standard Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Explain Question number 1 Adapted from N/A

Question

Explain why any integer can be written in the form 4k or 4k+1 or 4k+2 or 4k+3, where kZ.

[2]
a.

Hence prove that the square of any integer can be written in the form 4t or 4t+1, where tZ+.

[6]
b.

Markscheme

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 kZ      AG

[2 marks]

a.

(4k)2=16k2=4t        M1A1

(4k+1)2=16k2+8k+1=4t+1        M1A1

(4k+2)2=16k2+16k+4=4t      A1

(4k+3)2=16k2+24k+9=4t+1      A1

Hence, the square of any integer can be written in the form 4t or 4t+1, where tZ+.      AG

[6 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 1—Number and algebra » SL 1.6—Simple proof
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Topic 1—Number and algebra

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