Question

Problem 5. Find all x in Z that satisfy both of the following at the same time: x = 3 (mod 101) x =1 (mod 97)

Answer 1

theorm. chinese Remainder = a, (modmi) nea (modoma) 12 qr(mod my Les mom ., my denote & Positive integers that are relatively prime in pairs, and let aga..., a denote any r integers, then the e congruence can have common solutions. if 4 is one such solution, then an integer a satisfie es the congurance earm if and only ir nes of the form x=0+km for some integer k, Here mam.mm.... mr.

Comman son of the buth tongurance ma Find 133 (mod 101) = I (modet) by useing. chieniese reminder theorm to find comman son of the burb congurance. m, =l0l, m_=97, m= mom = 101897 9,=3, q=1 Find bed be o meni) b; = 1 (mod m;) tejsa I for jaz TUIs be = 1 (med mo leaba I (mod me) 976, al (mod love lokj (mo97) ☺ to to find ged of 97 a101 by useing extended euclid Algorithum lde (1097 + 4 97 = (24)4 +1 » 1 - 97- (24) 4 = 97- (24) [101- (1797) 1 5424).101 +(23)97 - eq" Takeing modulo 10) & gq respectively. 1=(28) (1) (mod 101 101 (-24) 31 (mod 91) 97(23731 imod 1012 | S

Camparing o a@ respectively we get 24 - واد يه = ط * = 91x( 25)x 3 + (12)x(-2) (1) را را با ما را با بابا بابا بابا رانا = X 2 24 = ي421 X = 485 Next Find all son of given tongurance system is (12و ۶ (۰) ۳۰۵) .8 = X » * = 4851 (mod 9797) gives all comman sonoF congurance system.