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Panduan Pemula Belajar Aritmetika Modular

Written By Nick Wednesday, September 26, 2018 Edit

belajar matematika di Hari PaskahDefenisi:
Misalkan $n$ adalah suatu bilangan bulat positif, $a$ dan $b$ adalah suatu bilangan bulat.
$a$ dikatakan kongruen $b$ modulo $n$, ditulis $a \equiv b\ (mod\ n)$ jika dan hanya jika $(a-b)$ adalah kelipatan $n$.Contoh 1:Alternatif Pembahasan: show

$3 \equiv 24\ (mod\ 7)$
Benar, karena $3 ΓÇô 24 = -21$ kelipatan dari $7$

Alternatif Pembahasan: show

$-31 \equiv 11\ (mod\ 7)$
Benar, karena $ΓÇô31 ΓÇô 11 = -42$ kelipatan dari $7$

Alternatif Pembahasan: show

$-15 \equiv -64\ (mod\ 7)$
Benar, karena $ΓÇô15 + 64 = 49$ kelipatan dari $7$

Alternatif Pembahasan: show

$13 \equiv -1\ (mod\ 7)$
Benar, karena $13 + 1 = 14$ kelipatan dari $7$

Alternatif Pembahasan: show

$23 \equiv 3\ (mod\ 7)$
Salah, karena $23 ΓÇô 3 = 20$ bukan kelipatan dari $7$

Contoh 2:Alternatif Pembahasan: show

$x \equiv 1\ (mod\ 10)$ jika dan hanya jika $x-1=10k$ Untuk setiap $k$ bilangan bulat.
Jika $k = 0, 1, 2, 3, ΓÇª$ maka berturut-turut $x = 1, 11, 21, 31, ...$

Begitu pula $k = -0, -1, -2, -3,...$ maka berturut-turut $x = -9, -200, 21, 31,...$
Begitu pula $k = -1, -2, -3,...$ maka berturut-turut $x = -9, -19, -29,...$

Dua barisan tersebut digabungkan sehingga himpunan penyelesaian $x \equiv 1\ (mod\ 10)$ adalah
$..., -29, -19, -9, 1, 11, 21, 31,...$

Sifat 1Alternatif Pembahasan: show

Untuk $a$ bilangan bulat sebarang dan $n$ suatu bilangan bulat positif berlaku $a ΓÇô a = 0 \cdot n $.
Dengan demikian $a \equiv a\ (mod\ n)$.

Alternatif Pembahasan: show

$a \equiv b\ (mod\ n)$
Ada $k$ suatu bilangan bulat.
Akibatnya
$b-a=-(a-b)$
$b-a=-(kn)$
$b-a=(-k)n$
Karena $-k$ juga suatu bilangan bulat, $b \equiv a\ (mod\ n)$.

Alternatif Pembahasan: show

$a \equiv b\ (mod\ n)$ dan $b \equiv c\ (mod\ n)$
Ada $h$ dan $k$ suatu bilangan bulat sehingga $a-b=hn$ dan $b-c=kn$.
Akibatnya:
$a-c=(a-b)+(b-c)$
$a-c=hn+kn$
$a-c=n(h+k)$
Karena $h+k$ juga bilangan, $a \equiv c\ (mod\ n)$.

Alternatif Pembahasan: show

$a \equiv b\ (mod\ n)$ dan $c \equiv d\ (mod\ n)$
Ada $h$ dan $k$ bilangan bulat sehingga $a-b=hn$ dan $b-c=kn$.
$(a+c)-(b+d)=(a-b)+(c-d)$
$(a+c)-(b+d)=hn+kn$
$(a+c)-(b+d)=n(h+k)$
Karena $h+k$ juga bilangan bulat, $a+c \equiv b+d\ (mod\ n)$.

Alternatif Pembahasan: show

$a \equiv b\ (mod\ n)$ dan $c \equiv d\ (mod\ n)$
Pandang
$ac=(b+hn)(d+kn)
$ac=bd+(bk+dh+hkn)n$
Karena $(bk+dh+hkn)$ bilangan bulat,
Ada $h$ dan $k$ bilangan bulat, $ac \equiv bd\ (mod\ n)$.

Alternatif Pembahasan: show

$a \equiv b\ (mod\ n)$
Ada $h$ bilangan sehingga $a-b=hn$
Karena $(a+c)-(b+c)=a-b=hn$
dengan demikian $a+c \equiv b+c\ (mod\ n)$.

Alternatif Pembahasan: show

$a \equiv b\ (mod\ n)$
Ada $h$ bilangan bulat sehingga $a ΓÇô b = hn $.
$ac-bc=(a-b)c=hnc=(hc)n$
Karena $hc$ bilangan bulat,
dengan demikian $ac \equiv bc\ (mod\ n)$.

Alternatif Pembahasan: show

Untuk bukti ini ita gunakan Induksi Matematika.
Untuk $k=1$, berlaku $a \equiv a\ (mod\ n)$.
Kita asumsikan $a^k \equiv b^k\ (mod\ n)$ berlaku,
Akan ditunjukkan $a^{k+1} \equiv b^{k+1}\ (mod\ n)$ juga berlaku.

Dari sifat [5], jika $a \equiv b\ (mod\ n)$ dan $c \equiv d\ (mod\ n)$ maka $ac \equiv bd\ (mod\ n)$
kita ganti $c$ oleh $a^k$ dan $d$ oleh $b^k$ diperoleh;
$aa^k \equiv bb^k (mod\ n)$ atau $a^{k+1} \equiv b^{k+1} (mod\ n)$

Contoh 3:Alternatif Pembahasan: show

Tampaknya kalkulator tidak dapat digunakan untuk menemukan jawaban atas masalah yang diajukan. Untuk itu kita gunakan cara lain untuk menyelesaikan masalah ini.

Kita tahu bahwa suatu bilangan bulat positif dibagi oleh 5 mempunyai sisa 0, 1, 2, 3, atau 4.
Penggunaan aritmetika modular akan membantu kita jika kita dapat menemukan bilangan bulat terkecil yang ekuivalen dengan pangkat dari 3, dan penggunaan sifat [7] dan [8] untuk membangun $3^{100}$ dan menemukan ekuivalensi mod 5.

Kita tahu bahwa $3^{2} \equiv 4\ (mod\ 5)$
Dengan demikian,
$3^3 \equiv 3 \cdot 4 \equiv 2\ (mod\ 5)$
$3^4 \equiv 3 \cdot 2 \equiv 1\ (mod\ 5)$

Dengan menggunakan sifat [8] kita peroleh,
$(3^4)^{25} \equiv 1^{25} (mod\ 5)$, atau
$3^{100} \equiv 1 (mod\ 5)$

Jadi: $3^{100}$ dibagi oleh $5$ mempunyai sisa $1$.

Sifat 2Alternatif Pembahasan: show

Karena $ca \equiv cb\ (mod\ n)$,
$c(a ΓÇô b) = ca ΓÇô cb = kn$ untuk suatu bilangan bulat $k$.

Kita tahu bahwa $d = FPB(c , n)$, dengan demikian, ada bilangan bulat saling prima [relative prime] $r$ dan $s$ yang memenuhi $c = dr$, $n = ds$.

Jika hasil ini kita substitusikan ke persamaan $c(a ΓÇô b) = ca ΓÇô cb = kn$ maka kita peroleh $r(a ΓÇô b) = ks$.

Hasil ini menunjukkan $s \mid r(a ΓÇô b)$, dan karena $FPB (r , s) = 1$, diperoleh $s (a ΓÇô b)$. Dengan kata lain,
$ca \equiv cb\ (mod\ n)$ maka $a \equiv b\ (mod\ n/d )$

Sifat 3Sifat 3sifat 2Sifat 4Alternatif Pembahasan: show

Kondisi $p \nmid c$ dan $p$ adalah bilangan prima ini mengakibatkan $FPB\ (c,p)=1$

Contoh 4:

Perhatikan $33 \equiv 15\ (mod\ 9)$, atau $3 \cdot 11 \equiv 3 \cdot 5\ (mod\ 9)$.
Karena $FPB(3,9)=1$ mengakibatkan $11 \equiv 5\ (mod\ 9)$

Perhatikan $-35 \equiv 45\ (mod\ 8)$, atau $5 \cdot (-7) \equiv 5 \cdot 9\ (mod\ 8)$.
Karena $5$ dan $8$ bilangan bulat saling prima mengakibatkan $-7 \equiv 9\ (mod\ 8)$

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