Teorija brojeva

Rezultati kolokvija održanog 30.6.2008.

Ime i prezime grupa Zad1 Zad2 Zad3 Zad4 Zad5 Zad6 2. kol. 1. kol. dod. Ukupno Ocjena
Ivan Andrić - 0 0 0 0 0 0 0 30 0 30  
Ivan Androš D 0 0 1 7 10 10 28 23 15* 66 2
Valentino Antolković B 10 10 5 10 10 6 51 48 20* 119 5
Stanko Arbutina - 0 0 0 0 0 0 0 0 0 0  
Petar Babić D 10 10 0 10 9 7 46 47 0 93  
Filip Bačić A 10 10 2 8 10 10 50 45 0 95  
Mladen Badrov B 0 7 0 10 10 10 37 42 20* 99 4
Kristina Bedeniković B 10 10 5 10 10 10 55 51 15* 121 5
Marko Begić A 1 7 0 10 6 10 34 41 0 75  
Inga Berghaus D 10 10 0 6 6 0 32 0 0 32  
Đurđica Blažević D 7 2 0 10 0 1 20 18 0 38  
Marina Borovnjak - 0 0 0 0 0 0 0 0 0 0  
Andrea Bosak B 10 10 5 10 9 10 54 57 15* 126 5
Marija Bošnjak D 5 10 4 10 9 10 48 54 0 102  
Dubravka Božić B 10 10 1 10 10 10 51 49 5 105  
Jana Buzolić C 10 3 5 10 2 3 33 47 0 80  
Ivana Ciganović C 10 10 4 9 7 5 45 50 15* 110 4
Krešimir Ciprić A 10 10 5 10 10 10 55 54 0 109  
Ivana Colarić C 10 10 0 10 10 10 50 55 15* 120 5
Josip Cvitanović - 0 0 0 0 0 0 0 33 0 33  
Irena Dabić - 0 0 0 0 0 0 0 7 0 7  
Martina Dorotek A 10 10 0 10 5 10 45 29 0 74  
Martina Doz C 10 10 0 5 10 10 45 35 10 90  
Ivana Duić A 6 10 4 10 5 7 42 50 5 97  
Mirna Erceg C 10 10 4 10 10 10 54 48 20* 122 5
Tamara Falak C 10 10 0 5 8 0 33 34 0 67  
Ana-Marija Flegar C 10 10 7 10 10 10 57 57 20* 134 5
Iva Gaćina D 10 10 0 10 10 10 50 42 15* 107 4
Kristina Gašpić - 0 0 0 0 0 0 0 0 0 0  
Tomislav Glavaš - 0 0 0 0 0 0 0 0 0 0  
Stjepan Glibo B 10 3 0 5 1 3 22 31 0 53  
Tena Gostović D 10 10 2 6 10 10 48 53 0 101  
Irena Grbac - 0 0 0 0 0 0 0 23 0 23  
Marko Gredičak B 0 8 0 10 10 10 38 34 0 72  
Katarina Gudelj - 0 0 0 0 0 0 0 0 0 0  
Ante Gunjača D 8 10 0 10 10 7 45 51 0 96  
Ivana Gut - 0 0 0 0 0 0 0 0 0 0  
Adnan Hamzabegović B 7 10 5 10 10 10 52 52 15* 119 5
Marijan Han D 8 10 5 10 9 10 52 30 0 82  
Slađana Herceg - 0 0 0 0 0 0 0 0 0 0  
Ksenija Hodak B 10 10 5 10 10 10 55 56 15* 126 5
Saša Ivanović - 0 0 0 0 0 0 0 0 0 0  
Renata Jelić C 10 10 0 10 8 7 45 58 20* 123 5
Marcela Jusup D 0 1 0 8 3 6 18 23 0 41  
Matej Kadvolt - 0 0 0 0 0 0 0 0 0 0  
Anđelka Kalfić A 10 10 1 7 9 10 47 28 5 80  
Nena Kekez B 10 10 1 10 10 10 51 46 5 102  
Ana Keller D 10 10 0 7 4 1 32 29 0 61  
Stevo Kentrić C 0 6 0 8 10 1 25 25 0 50  
Kristina Knežević A 10 8 0 5 1 6 30 22 0 52  
Iva Kodrnja C 10 10 7 10 10 10 57 60 0 117  
Vedran Kogej A 10 10 0 9 10 10 49 57 15* 121 5
Valentina Koretić B 2 10 5 10 2 5 34 0 15* 49  
Maja Kovač D 10 10 5 9 9 2 45 36 0 81  
Vedran Krnjaić - 0 0 0 0 0 0 0 0 0 0  
Josipa Kunović A 10 10 0 10 5 6 41 40 5 86  
Nenad Kuren - 0 0 0 0 0 0 0 0 0 0  
Danijel Laklija B 6 10 0 10 10 10 46 60 5 111  
Tomislav Lektorić - 0 0 0 0 0 0 0 0 0 0  
Marija Lijović - 0 0 0 0 0 0 0 0 0 0  
Ivana Major Šomođi D 1 7 0 3 7 0 18 34 5 57  
Barbara Malkoč C 7 10 0 10 10 7 44 21 5 70  
Anamaria Mandac A 10 8 0 10 8 6 42 58 15* 115 4
Marko Marić C 1 2 5 10 10 10 38 38 10 86  
Zoran Marinković B 10 2 0 8 0 0 20 0 0 20  
Marina Marjanović-Zulim - 0 0 0 0 0 0 0 43 0 43  
Goran Matijević D 10 7 0 10 9 10 46 39 5 90  
Jasmin Matijević D 10 10 5 8 10 8 51 46 5 102  
Hani Matulić B 2 10 0 10 10 10 42 46 15* 103 4
Kristina Micak - 0 0 0 0 0 0 0 0 0
Mia Mijatović A 10 10 5 6 10 10 51 54 0 105  
Martina Milaković C 3 10 0 10 2 2 27 32 10 69  
Vesna Milenković - 0 0 0 0 0 0 0 0 0 0  
Jelena Milojević D 10 10 0 10 10 10 50 48 15* 113 4
Daniel Miočić B 10 10 8 10 10 10 58 57 0 115  
Nikolina Morić C 10 10 2 10 10 10 52 48 0 100  
Ivan Mušicki A 10 10 0 10 3 3 36 47 15* 98 4
Josipa Nikolić C 10 1 0 10 6 3 30 58 10 98  
Mia Nižić D 10 10 2 7 9 2 40 55 10 105
Adnana Nurikić B 10 10 0 10 10 6 46 60 20* 126 5
Andrea Ostojić A 10 10 0 10 10 10 50 52 20* 122 5
Mia Pamuković B 10 5 10 10 10 10 55 58 10 123  
Jelena Patarčić A 10 10 10 10 5 10 55 47 0 102  
Lana Periša D 6 10 5 10 9 10 50 0 0 50  
Mihael Petrić A 10 10 6 10 10 10 56 52 15* 123 5
Ivona Petrović D 6 10 1 8 10 7 42 54 15* 111 4
Tina Petrović D 10 10 5 10 10 10 55 53 15* 123 5
Luka Podrug C 10 10 0 10 10 10 50 0 10 60  
Igor Posavec C 10 10 0 10 10 10 50 55 0 105  
Bojan Poturica B 10 10 5 10 4 10 49 0 0 49  
Ana Prskalo C 5 8 4 7 8 3 35 47 10 92  
Ana Radeljak C 10 10 4 7 10 2 43 41 20 104 4
Iva Radoš A 10 8 0 10 10 6 44 33 0 77  
Vedran Rafaj C 10 10 5 10 10 10 55 45 10 110  
Maja Rod A 10 6 0 8 7 10 41 35 0 76  
Ivana Rogulj A 10 10 0 9 9 10 48 43 0 91  
Maja Rogulja D 2 10 5 10 4 10 41 50 5 96  
Vedran Rumenović D 1 0 5 9 7 3 25 0 0 25  
Dean Singer B 10 10 10 10 10 10 10 39 20* 119 5
Anja Skendžić A 3 10 0 8 7 5 33 48 0 81  
Ana Soldo - 0 0 0 0 0 0 0 0 0 0  
Bernardica Solina A 3 10 8 6 5 10 42 33 5 80  
Alenka Solje D 10 10 1 10 9 10 50 54 10 114  
Denis Stažnik A 0 3 5 10 10 10 38 47 0 85  
Željka Stojanac B 10 10 10 10 10 10 60 58 15* 133 5
Sanja Stojanović A 10 10 0 10 9 10 49 38 20* 107 4
Maja Šarić C 10 10 1 10 10 10 51 48 20* 119 5
Marijana Šikić B 10 10 0 10 8 10 48 51 0 99  
Nevenka Šistov B 10 10 0 5 6 0 31 0 0 31  
Kristina Šoković D 10 10 0 10 4 5 39 35 0 74  
Lana Špiljarić D 10 1 6 9 8 10 44 48 0 92  
Mate Šunjić A 10 10 10 10 10 10 60 60 10 130  
Matija Švenda B 0 0 5 4 10 0 19 36 10 65  
Višnja Tenčić A 10 10 5 10 10 10 55 60 0 115  
Ilija Tomaš B 10 10 0 10 10 10 50 54 0 104  
Vinka Trevižan - 0 0 0 0 0 0 0 12 5 17  
Tihana Trivić A 10 10 0 10 9 8 47 39 20* 106 4
Ivana Turčić C 10 10 3 10 10 10 53 46 20* 119 5
Kristina Ugarković D 10 10 1 10 10 10 51 57 0 108  
Martina Ursa - 0 0 0 0 0 0 0 0 0 0  
Dominik Vidušin A 0 8 0 10 9 6 33 0 0 33  
Dragutin Viher C 0 7 4 10 10 10 41 53 0 94  
Marija Visković B 8 10 7 10 6 9 50 55 10 115  
Ante Višić - 0 0 0 0 0 0 0 0 0 0  
Dolores Vlašić A 10 10 0 10 6 10 46 56 0 102  
Ana Vranešić C 10 10 0 9 10 4 43 37 5 85  
Denis Vrdoljak C 10 10 0 9 8 6 43 38 20* 101 4
Mirta Vučković - 0 0 0 0 0 0 0 0 0 0  
Martina Vukelić B 10 10 0 10 10 10 50 58 20* 128 5
Bernard Živković B 0 10 5 10 4 10 39 46 10 95  
Ana Žoldin C 10 10 1 10 10 3 44 44 15* 103 4

Zadaće se mogu pogledati u ponedjeljak, 7.7.2008. u 15:15.

Rješenja zadataka:

grupa A
1. 5x2 + 3xy + 7y2
2. h(-103) = 5;   x2 + xy + 26y2, 2x2 ± xy + 13y2, 4x2 ± 3xy + 7y2
3. Ako su m i n relativno prosti, onda je s(mn) = ∏p|mn p = ∏p|m p · ∏p|n p = s(m)s(n), pa je funkcija s multiplikativna.
    Označimo zadanu sumu sa g(n). Funkcija g je multiplikativna, pa je g(n) = ∏p|n (μ(1)σ(1)+μ(p)σ(p)) = ∏p|n (-p) = (-1)ω(n) s(n).
4. 543/193 = [2,1,4,2,1,3,3];   korijen298 = [17,3,1,4,5,1,1,5,4,1,3,34]
5. (21,220,221), (85,204,221), (104,195,221), (140,171,221), (60,221,229), (221,1428,1445), (221,1872,1885), (221,24420,24421)
6. (145,12), (42049,3480)

grupa B
1. 2x2 - xy + 16y2
2. h(-107) = 3;   x2 + xy + 27y2, 3x2 ± xy + 9y2
3. Ako su m i n relativno prosti, onda je λ(mn) = (-1)ω(m)+ω(n) = (-1)ω(m) (-1)ω(n) = λ(m)λ(n), pa je funkcija λ multiplikativna.
    Označimo zadanu sumu sa g(n). Funkcija g je multiplikativna, pa je g(n) = ∏p|n (μ(1)λ(1)+μ(p)λ(p)) = ∏p|n (1+1) = 2ω(n).
4. 654/173 = [3,1,3,1,1,4,4];   korijen292 = [17,11,2,1,3,8,3,1,2,11,34]
5. (152,285,323), (36,323,325), (323,2736,2755), (323,3060,3077), (323,52164,52165)
6. (199,15), (79201,5970)

grupa C
1. 3x2 + xy + 9y2
2. h(-127) = 5;   x2 + xy + 32y2, 2x2 ± xy + 16y2, 4x2 ± xy + 8y2
3. Funkcija ω nije multiplikativna, jer je npr. ω(2 · 3) = 2, a ω(2) · ω(3) = 1 · 1 = 1.
    Označimo zadanu sumu sa g(n). Funkcija g je multiplikativna, pa je g(n) = ∏p|n (μ(1)τ(1)+μ(p)τ(p)) = ∏p|n (1-2) = (-1)ω(n).
4. 765/179 = [4,3,1,1,1,7,2];   korijen295 = [17,5,1,2,3,2,6,3,2,1,5,34]
5. (115,276,299), (180,299,349), (299,1932,1955), (299,3432,3445), (299,44700,44701)
6. (143,12), (40897,3432)

grupa D
1. 4x2 + 3xy + 7y2
2. h(-131) = 5;   x2 + xy + 33y2, 3x2 ± xy + 11y2, 5x2 ± 3xy + 7y2
3. Ako su m i n relativno prosti, onda je f(mn) = 2ω(m)+ω(n) = 2ω(m) 2ω(n) = f(m)f(n), pa je funkcija f multiplikativna.
    Označimo zadanu sumu sa g(n). Funkcija g je multiplikativna, jer je |μ(d)| = μ(d) · μ(d), pa je g(n) = ∏p|n (|μ(1)|+|μ(p)|) = ∏p|n (1+1) = 2ω(n).
4. 876/191 = [4,1,1,2,2,1,1,6];   korijen286 = [16,1,10,3,3,2,3,3,10,1,32]
5. (135,352,377), (145,348,377), (152,345,377), (260,273,377), (336,377,505), (377,2436,2465), (377,5460,5473), (377,71064,71065)
6. (168,13), (56447,4368)

Andrej Dujella