User interface language: English | Español

Date May 2022 Marks available 2 Reference code 22M.1.SL.TZ2.3
Level Standard Level Paper Paper 1 (without calculator) Time zone Time zone 2
Command term Prove Question number 3 Adapted from N/A

Question

Consider any three consecutive integers, n-1, n and n+1.

Prove that the sum of these three integers is always divisible by 3.

[2]
a.

Prove that the sum of the squares of these three integers is never divisible by 3.

[4]
b.

Markscheme

n-1+n+n+1          (A1)

=3n           A1

which is always divisible by 3           AG

 

[2 marks]

a.

n-12+n2+n+12   =n2-2n+1+n2+n2+2n+1           A1

attempts to expand either n-12 or n+12    (do not accept n2-1 or n2+1)          (M1)

=3n2+2           A1

demonstrating recognition that 2 is not divisible by 3 or 23 seen after correct expression divided by 3            R1

 

3n2 is divisible by 3 and so 3n2+2 is never divisible by 3

OR  the first term is divisible by 3, the second is not

OR  3n2+23  OR  3n2+23=n2+23

hence the sum of the squares is never divisible by 3          AG

 

[4 marks]

b.

Examiners report

Most candidates were able to earn full marks in part (a), though some were not able to provide the required reasoning to earn full marks in part (b). In many cases, candidates did not seem to understand the nature of a general deductive proof, and instead substituted different consecutive integers (such as 1, 2,3 ), showing the desired result for these specific values, rather than an algebraic generalization for any three consecutive integers.

a.
[N/A]
b.

Syllabus sections

Topic 1—Number and algebra » SL 1.6—Simple proof
Show 41 related questions
Topic 1—Number and algebra

View options