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Date May 2022 Marks available 6 Reference code 22M.1.AHL.TZ1.8
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 1
Command term Prove Question number 8 Adapted from N/A

Question

Consider integers a and b such that a2+b2 is exactly divisible by 4. Prove by contradiction that a and b cannot both be odd.

Markscheme

Assume that a and b are both odd.             M1


Note: Award M0 for statements such as “let a and b be both odd”.
Note: Subsequent marks after this M1 are independent of this mark and can be awarded.


Then a=2m+1 and b=2n+1            A1

a2+b22m+12+2n+12

=4m2+4m+1+4n2+4n+1            A1

=4m2+m+n2+n+2            (A1)

(4m2+m+n2+n is always divisible by 4) but 2 is not divisible by 4. (or equivalent)            R1

a2+b2 is not divisible by 4, a contradiction. (or equivalent)            R1

hence a and b cannot both be odd.            AG


Note: Award a maximum of M1A0A0(A0)R1R1 for considering identical or two consecutive odd numbers for a and b.

 

[6 marks]

Examiners report

Most candidates did not present their proof in a formal manner and merely relied on an algebraic approach rendering the proof incomplete. Very few candidates earned the first mark for making a clear assumption that a and b are both odd. A significant number of candidates only considered consecutive or identical odd numbers. The required reasoning to complete the proof were often poorly expressed or missing altogether. Only a small number of candidates were awarded all the available marks for this question.

Syllabus sections

Topic 1—Number and algebra » AHL 1.15—Proof by induction, contradiction, counterexamples
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Topic 1—Number and algebra

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