Elementarna teorija brojeva

Rezultati kolokvija održanog 4.7.2008.

Ime i prezime grupa Zad1 Zad2 Zad3 Zad4 Zad5 Zad6 2. kol. 1. kol. zad. dod. Ukupno Ocjena
Vanja Andrašić B 10 10 10 10 10 0 50 60 20 5 135  
Martina Arapović C 10 10 10 10 10 7 57 49 20 10* 136 5
Jelena Artuković - 0 0 0 0 0 0 0 0 0 0 0  
Vesna Bajs C 2 10 2 1 0 0 15 33 10 0 58  
Mateja Bolšec B 8 10 10 10 8 0 46 46 17 10* 119 4
Andrea Brajčić C 5 10 2 10 10 10 47 48 20 10* 125 4
Ružica Brdarić B 10 10 8 10 10 0 48 31 20 0 99  
Anita Brezina C 6 10 6 1 6 0 29 33 20 0 82  
Valentina Bubnjar - 0 0 0 0 0 0 0 0 0 0 0  
Danijela Cvitanović A 10 10 3 9 2 0 34 38 20 0 92  
Mirela Deranja - 0 0 0 0 0 0 0 0 0 0 0  
Leonardo Đaković A 10 10 10 10 10 8 58 57 20 20* 155 5
Paula Faist A 5 10 10 8 2 0 35 48 20 0 103  
Mario Fanjak D 10 10 10 10 10 10 60 50 20 10* 140 5
Helena Frančešević C 10 10 8 9 1 5 43 36 20 0 99  
Marko Fruk A 9 10 10 10 10 5 54 34 20 0 108  
Marija Gluhak D 10 10 10 10 7 2 49 36 20 10* 115 4
Sanda Golub C 10 10 10 10 10 5 55 54 20 10* 139 5
Mateja Goričanec B 10 10 10 10 10 0 50 59 20 10* 139 5
Marija Gudelj A 3 10 10 10 8 0 41 40 20 0 101  
Hrvoje Held B 8 10 10 8 10 0 46 35 18 0 99  
Zvjezdana Hrešć C 10 10 2 10 10 1 43 53 20 5 121  
Marija Imrović C 3 10 10 10 10 10 53 38 18 0 109  
Elvira Jakčin D 0 10 2 9 3 0 24 31 18 0 73  
Željka Jakuš-Mejarec D 10 10 10 10 10 5 55 55 20 0 130  
Lidija Jokoš D 10 10 8 9 1 3 41 32 20 0 93  
Josipa Jurić A 10 10 10 10 10 4 54 46 20 0 120  
Miran Kapelac B 0 10 10 10 6 0 36 40 19 0 95  
Nikolina Knežević C 10 10 10 10 10 10 60 55 20 10* 145 5
Nino Koncul - 0 0 0 0 0 0 0 0 0 0 0  
Martina Kontrec C 5 10 10 10 7 0 42 34 20 0 96  
Jelena Kostić B 1 10 10 10 6 0 37 24 20 0 81  
Ivana Kožić - 0 0 0 0 0 0 0 0 0 0 0  
Davor Kucelj A 5 10 10 10 10 0 45 30 18 0 93  
Romana Kušić C 4 10 10 10 0 0 34 30 16 0 80  
Željka Liović A 10 10 10 10 10 5 55 59 20 15* 149 5
Petra Lončar B 10 10 10 10 10 8 58 60 20 5 143  
Radomir Lončarević C 10 10 10 10 10 10 60 60 20 10* 150 5
Tihana Mačković D 4 10 10 10 9 3 46 44 20 0 110  
Ana Marić A 10 10 10 9 3 0 42 39 16 0 97  
Rozalija Marko D 10 10 10 10 10 7 57 45 10 0 112  
Daniel Marković C 10 10 10 10 10 0 50 60 18 0 128  
Lidija Matić D 10 10 10 10 10 0 50 50 20 0 120  
Zlatko Matijašević D 10 10 10 10 10 3 53 51 20 5 129  
Janja Mikić - 0 0 0 0 0 0 0 0 0 0 0  
Vesna Mikulčić C 10 10 2 10 10 0 42 39 20 0 101  
Tamara Mlinar - 0 0 0 0 0 0 0 0 0 0 0  
Miro Ojvan B 10 10 8 10 10 4 52 44 20 20* 136 5
Ivana Oršolić B 10 10 8 10 8 2 48 57 19 5 129  
Marija Perić D 3 10 10 10 10 5 48 23 20 0 91  
Marijana Pili A 10 10 10 10 10 3 53 42 19 0 114  
Ivan Piskač B 8 10 10 10 10 0 48 51 20 0 119  
Marijana Piškor D 9 10 8 9 2 0 38 35 15 0 88  
Snježana Plantak A 10 10 10 10 10 10 60 59 20 5 144  
Marina Pleša B 10 10 10 10 10 0 50 59 20 0 129  
Maja Poljak A 3 10 10 10 10 0 43 25 10 0 73  
Ivana Ptiček A 10 10 8 10 8 0 46 39 18 5 108  
Petar Radanović A 10 10 10 10 10 10 60 43 19 10* 132 4
Magdalena Radočaj C 10 10 10 10 10 10 60 49 20 10* 139 5
Petra Raguž D 10 10 10 10 10 3 53 59 20 5 137  
Mihaela Rajič C 10 10 8 10 4 7 49 48 20 5 122  
Jelena Rakić D 10 10 10 10 8 7 55 45 19 0 119  
Marijan Ralašić C 10 10 10 10 10 6 56 59 20 0 135  
Maja Ramljak A 10 10 10 10 3 1 44 47 20 0 111  
Dean Raonić - 0 0 0 0 0 0 0 0 0 0 0  
Iva Rešetar B 10 10 10 10 7 0 47 33 20 0 100  
Ines Rutić C 9 10 10 8 10 7 54 40 19 0 113  
Milena Savić A 10 10 8 10 10 4 52 52 20 5 129  
Tomislav Sente - 0 0 0 0 0 0 0 0 0 0 0  
Kristina Serec B 10 10 10 10 10 0 50 48 20 0 118  
Dinko Smolčić B 10 10 10 10 10 0 50 47 18 0 115
Filip Srića A 0 0 0 0 0 2 2 0 9 0 11  
Tatjana Stojanović A 10 10 10 10 10 5 55 44 10 0 109  
Manuela Šimek B 10 10 10 10 10 3 53 53 20 10* 136 5
Amadea Šimić A 10 10 10 10 10 0 50 40 20 10* 120 4
Kristina Šiško D 0 10 10 10 10 0 40 40 20 0 100  
Maja Škrlec D 10 10 10 10 10 10 60 47 20 10* 137 5
Marija Štimac A 10 10 10 10 10 7 57 46 20 15* 138 5
Branka Tišljarić - 0 0 0 0 0 0 0 25 9 0 34  
Marko Tonković A 10 10 10 10 10 0 50 35 20 0 105  
Tomislav Trabalko - 0 0 0 0 0 0 0 19 10 0 29  
Maja Vauhner B 10 10 10 10 8 0 48 43 17 0 108  
Igor Vidović B 10 10 10 10 10 0 50 36 20 5 111  
Štefica Vitković D 10 10 10 9 10 2 51 39 20 5 115  
Sanja Vugrinec A 10 10 10 10 10 10 60 51 20 5 136  
Kristina Zeman D 10 10 10 10 10 3 53 46 20 0 119  
Nika Katerina Zidarić D 10 10 10 10 10 5 55 59 20 5 139  
Helena Županić B 10 10 10 10 10 0 50 41 10 0 101  

Zadaće se mogu pogledati u srijedu, 9.7.2008. u 14:15.

Rješenja zadataka:

grupa A
1. (37/547) = -1,   (38/547) = -1
2. 346
3. (57,76,95), (95,168,193), (95,228,247), (95,900,905), (95,4512,4513)
4. √76 = [8,1,2,1,1,5,4,5,1,1,2,1,16]
5. (49,5), (4801,490)
6. Ako je (n-2)2+(n-1)2+ n2+(n+1)2+(n+2)2 = 5(n2+2) kvadrat, onda 5|n2+2, tj. n2 ≡ 3 (mod 5). Ali kvadrati pri dijeljenju sa 5 daju ostatke 0, 1 ili 4.

grupa B
1. (42/659) = 1,   (43/659) = -1
2. 395
3. (35,84,91), (60,91,109), (91,312,325), (91,588,595), (91,4140,4141)
4. √67 = [8,5,2,1,1,7,1,1,2,5,16]
5. (39,4), (3041,312)
6. Ako bi postojalo cjelobrojno rješenje dane jednadžbe, onda bi vrijedilo 5x2 ≡ 1999 ≡ 3 (mod 4), odnosno x2 ≡ 3 (mod 4), što je nemoguće.

grupa C
1. (46/769) = -1,   (47/769) = 1
2. 445
3. (56,105,119), (119,120,169), (119,408,425), (119,1008,1015), (119, 7080,7081)
4. √86 = [9,3,1,1,1,8,1,1,1,3,18]
5. (55,6), (6049,660)
6. Neka je x = a2, y = b2, z = c2. Iz (a+b)(a-b)=96, (b+c)(b-c)=96, dobije se da je a = 14, b = 10, c = 2, tj. x = 196, y = 100, z = 4.

grupa D
1. (58/877) = -1,   (59/877) = 1
2. 495
3. (55,132,143), (24,143,145), (143,780,793), (143,924,935), (143,10224,10225)
4. √93 = [9,1,1,1,4,6,4,1,1,1,18]
5. (82,9)
6. Iz n2 + 4n - 5 = m2 slijedi (n+2)2 - m2 = (n+2-m)(n+2+m) = 1 · 9 = (-9) · (-1) = 3 · 3 = (-3) · (-3), odkud se dobiju rješenja n = 3, -7, 1, -5.

Andrej Dujella