Date | May 2011 | Marks available | 8 | Reference code | 11M.1.hl.TZ0.3 |
Level | HL only | Paper | 1 | Time zone | TZ0 |
Command term | List and Prove that | Question number | 3 | Adapted from | N/A |
Question
Prove that the number 14641 is the fourth power of an integer in any base greater than 6.
For a,b∈Z the relation aRb is defined if and only if ab=2k , k∈Z .
(i) Prove that R is an equivalence relation.
(ii) List the equivalence classes of R on the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
Markscheme
14641 (base a>6 ) =a4+4a3+6a2+4a+1 , M1A1
=(a+1)4 A1
this is the fourth power of an integer AG
[3 marks]
(i) aRa since aa=1=20 , hence R is reflexive A1
aRb⇒ab=2k⇒ba=2−k⇒bRa
so R is symmetric A1
aRb and bRc⇒ab=2m, m∈Z and bRc⇒bc=2n , n∈Z M1
⇒ab×bc=ac=2m+n , m+n∈Z A1
⇒aRc so transitive R1
hence R is an equivalence relation AG
(ii) equivalence classes are {1, 2, 4, 8} , {3, 6} , {5, 10} , {7} , {9} A3
Note: Award A2 if one class missing, A1 if two classes missing, A0 if three or more classes missing.
[8 marks]
Examiners report
This was not difficult but a surprising number of candidates were unable to do it. Care with notation and logic were lacking.
The question was at first straightforward but some candidates mixed up the properties of an equivalence relation with those of a group. The idea of an equivalence class is still not clearly understood by many candidates so that some were missing.