Introduction to Congruences

Bismillahirrahmanirrahim.

Assalamualaikum warahmatullahi wabarakatuh.

Dlm seminggu ni, rase mcm malas je nak mmbaca. Rasa mcm nak dengar ape yg lecturer ckp je. Hehe. Dari dulu mmg suka dgr ape yg lecturer ckp je, sbb fokus 100% in class mmg antara tips for excellence. Tpi bila dah masuk U ni, yup, mmg kene studi sndiri...;)

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Number Theory. Chapter 4: Congruences. (utk yg tak amik Math...kita still bleh berkongsi....jgn takut Math ea~huhu)

Definition:

Let m be a positive integer. If a and b are integers, we say that a is congruent to b modulo m if m | (a-b).

Kat sini dia nak introduce terms yg akan digunakan dlm bab congruences ni. Kira mcm dia baru nak introduce congruences la. Mmg kita tak paham lagi, ape kebende nye congruences ni? Tapi slepas Dr Pah ajar dlm kelas...

Cara nak tulis congruence ni: "If a is congruent to b modulo m, so we write a≡b (mod m). Klu m does not divides (a-b), we say a and b are incongruent. The integer m is called the modulus of the congruence. The plural is moduli.

See, introduction pasal congruences dlm buku Number Theory ni, dia explain pasal term2s yg ada dulu. Klu tak paham lagi stakat ni...takpe. Jgn risau. Kita tgok seterusnya, dia bg contoh.

Disebabkan kita dah biasa tgok bentuk divisibility, ataupun pembahagian (÷) so mesti kita dah biasa tgok bentuk ni kan?:

                                      m | (a-b)

Sekarang, dia introduce cara baru utk menulis divisibility di atas:

                                      a≡b (mod m)

kiranya mcm congruences ni ialah cara baru utk tulis m | (a-b) la...bleh la nak cakap mcmtu. Tapi congruence ni ada bnyak properties. Properties dia kita akan discover kemudian. Mula2 kita tgok beberapa contoh dulu ea:

                 (22-4)÷9 or  9 | (22-4)  -> ni same bentuk dgn m | (a-b)

                   22-4=18, so 18 can be divided by 9

                   Cara nak tulisnye dlm congruences: 22≡4 (mod 9)

Atau bahasa mudah dia...22 kalau bahagi 9, akan tinggal baki 4! Haha. See next example:

                   9 | (200-2) mcm bntuk m | (a-b)

                   200 – 2 = 198, 198 bleh dibahagi dgn 9

                   So, 200≡2 (mod 9)

                   Tapi, let see (13-5)÷9. 9 cannot be divided by (13-5) kan? Nanti jadi fraction 8/9

                   because 13 – 5 = 8

                   So kita takleh tulis diorang dalam bentuk congruence.

                   Kita cakap... 13 and 5 is incongruent in mod 8

Ok. Sekarang dah paham kan? Sekarang kita leh introduce salah satu property congruence. Takde la macam properties, tpi Theorem la, Theorem 4.1.

If a and b are integers, then a≡b (mod m) if and only if there is an integer k with km = a – b, so that a = b + km.

Err, paham tak? Tak? Takpe, kita tgok proof dia.

          Kalau a≡b (mod m),

          so m | (a-b).

          So berdasarkan Definition of Divisibility (tajuk awal2 Number Theory),

          there exist an integer k which, (a – b) = km.

          Ok. So, pindah2kan,

          a = b + km...dah dapat...

Haa...dah dapat? Proof selanjutnya bleh bca dalam buku ea ;)

OKlah, Sampai sini dulu.....dah ngantuk,nak tido....hehe...assalamualaikum.