Question

Problem 1 Use the Chinese remainder theorem, find all integers x such that: (20 pts) x = 1 (mod 5) x = 2 (mod 7) x = 3 (mod 9) x = 4 (mod 11)

Answer 1

Please find the solution below.
AB - Given: N=1 (mods) nda) R= 2( mod 7) x= 3 (mod 9) au (mod "1) Telor Applying chinese remainder theorem Since 5,7,9,11 are pairwise relatively prime, this Chinese Remaunder theorem tells us that there is a unique solution moduto in conere m= 5.7.9.11 = 3465 Hence, K=4 m=5, M-76m3-9, me = 11 AL=1 22=2, 933, a=4 to get the colution 2 m . Mam z mu-7.9.11 – 693. 21 m ZZE MIM2= MIMO mu=5.9.11 21495 MIMs = mime mu: 5.7.11=385 zumimum, Mg M 37 5.7.9 - 315 Yis z, mod (mi) = 693-1 mod (5) = 2 Y 2-2 mod me) = 45 mode) = 3 mod a=4 Y 3= 23 mod m3) = 385 mod 20 =8. Juhi had my)=315 wi=4,7 mod m) = 2.693 (mod 346) = 1386 w2=5222 (mod m) – 3.495 (mod 3465) - 1985 W32Y373 mod m) = 0.385 mod 3465) =1540 wu-Yu 24 hodin) = 8.315 (mod 3465)-2520 se cc

Hence, the solution is a=9,0, +92 W2 +93wzt Auwa mod in) = (-1386)+ 62-14854/ 3.4540)+ 4.2920) (mod 3465) = 1386+2970+46207 10080 ( mod 3665) =19,006 (3465) pobrino = 168) mod (3465) Makin MODE

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