5-mavzu. Qoldiqlar haqidagi Xitoy teroemasi. Ixtiyoriy modul bo`yicha n-darajali taqqoslamalar

3*. Berilgan modul bo’yicha har qanday butun son o’zining qoldig’i bilan taqqoslanishini isbot qiling.
4. Quyidagi taqqoslamalarni qanoatlantiradigan x ning barcha qiymatlarini toping:
a) x  0 (mod 3); b) x  1 (mod 2).
5. Quyidagi taqqoslamalarni qanoatlantiradigan m ning barcha qiymatlarini toping: 3r + 1  r + 1 (mod m).
6. Agar x = 13 soni x  5 (mod m) taqqoslamani qanoatlantirsa, modulning mumkin bo’lgan qiymatlarini toping.
7*. Agar n – toq son bo’lsa, u holda n² - 1  0 (mod 8) taqqoslama o’rinli ekanligini ko’rsating.
8*. Agar 100a + 10b + c  0 (mod 21) bo’lsa, u holda a – 2b + 4c  0 (mod 21) taqqoslamaning o’rinli ekanligini ko’rsating.
9. Agar 3ⁿ  -1 (mod 10) bo’lsa, u holda 3ⁿ⁺⁴  -1 (mod 10) (nN ) taqqoslamaning o’rinli ekanligini ko’rsating,.
10*. 2^{11 31}  2 (mod 1131) taqqoslamaning to’g’riligini ko’rsating.
11*. Agar x = 3n + 1, n = 0, 1, 2,.... bo’lsa, u holda 1 + 3^x + 9^x ning 13 ga bo’linishini ko’rsating.
12. N = 111823221319 soni 7 modul bo’yicha absolyut qiymati bo’yicha eng kichik qanday son bilan taqqoslanadi?
13. 3¹⁴  -1 (mod 29) ni tekshiring.
14. 1532⁵ – 1 ni 9 ga bo’lganda hosil bo’ladigan qoldiqni toping.
15*. Agar a  b (mod pⁿ) bo’lsa, u holda a^p  b^p (mod pⁿ⁺¹) ni isbotlang.
16 Agar ax  bx (mod m) bo’lsa, u holda ni isbotlang.
17*. Agar a₄a₃a₂a₁a₀  0 (mod 33) bo’lsa, u holda
a₄ + a₃a₂ + a₁a₀  0 (mod 33) ni isbotlang. a_i+1 = 0 da a_i+1a_i = a_i
deb oling.
18*. Berilgan sonning oxirgi ikkita raqamini toping: a) 9⁹ ; b) 7⁹ .
19*. r^r+2 + (r+2)^r  0 (mod 2r+2) taqqoslamani isbot qiling, bu yerda r > 2.