Kuis_1_Numbers-solScele.pdf

Kuis_1_Numbers-solScele.pdf - Quiz 1 Matematika Diskret...

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1 Quiz 1: Matematika Diskret 2 (kode soal: MD-Q2- 02/04/06/10 ) Semester Gasal 2015/2016 Fasilkom UI 1. Buktikan teorema berikut: Misalkan m bilangan bulat positif; a,b , dan c adalah bilangan bulat, jika ac bc (mod m ) dan gcd ( c,m ) = 1, maka a b (mod m ) Direct proof : ac bc (mod m ) dan gcd ( c,m ) = 1 ac bc (mod m ) m| ( ac bc ) m | c ( a b ) gcd( c,m ) = 1, artinya m tidak membagi habis c m |( a b ) a b (mod m ) Jika a = bq + r dan a,b,q dan r masing-masing adalah bilangan bulat, maka berlaku gcd( a,b ) = gcd( b,r ) (lihat slide kuliah halaman 63 66) Jika a dan b masing-masing adalah bilangan bulat, maka berlaku ab = gcd( a,b ).lcm( a,b ) Misalkan a dapat difaktorisasi ke dalam faktor-faktor prima Dan b dapat difaktorisasi ke dalam faktor-faktor prima ( )( ) ) ) ) ) ) ) ) ) ab = gcd( a,b ).lcm( a,b ) Untuk a dan b bilangan bulat, m bilangan bulat positif, berlaku (( a + b ) mod m = (( a mod m ) + ( b mod m )) mod m Berdasarkan definisi mod dan kongruensi, didapat a ( a mod m )(mod m ) dan b ( b mod m )(mod m ) sehingga ( a + b ) ( a mod m ) + ( b mod m )( mod m) dan (( a + b ) mod m = (( a mod m ) + ( b mod m ))

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