Question

Please do questions 1a, and 1b and 2d do not worry about doing any of the others. Please explain them as well i am trying to understand

1. a) Disprove that Va, b, c e Z, a + 0, if alc and b|c, then ab|c. b) Using the definition of divisibility, prove that Va, b, c e Z, a + 0, if alb, then aſbc. 2. Use modular arithmetic to calculate the following. Show your work, with all steps you needed. a) (7072 mod 23) · (3011 mod 23) mod 23 = b) Find an integer a so that a mod 29 = 18 and -58 < a < -29 c) (620197297 · 99918734) mod 5= d) 114537 mod 19 =

Answer 1

Solution i 7 Take a= 2 6 = 4, c=4 Clearly alc because 214 1 ble because 414 but able because ab=8 and (=h and 844. 6 let aib. ct 2 with ato Given that alb dherefore by definition of divisibility there exist mez such that b=am - bc = cam - bC= a (mc) = a1be7

Solukion 20 Therefore Here ged (11.19)=1 by Fermats theorem 19-1 and 19 is prime. we get Codral 238 Goud 19) Viel (mod 19) = -> (113 4522 = '1' El mod 19) — o Now iVe 18 (mod 19) beause 19 = 1331 - (1375 18" (med 13) = " 1889568 (mod 19) aus 18 (modig) because IPP 95683 lP (mod 19) From 0 and ③ we get 4522 15 "?= 1.18 mod 1g) - 4537 11 = 18 (modig) 3) 4537 I li mod ig = 18 Thank you