Teori Bilangan

Biar ga ngantuk, ni bisa dibaca (dongeng sebelum tidur lah..) ato malah ntar bakal mimpi buruk, ah semoga aja mimpi indah (mimpi ketemu saya -_-)

1. a.) Suatu bilangan habis dibagi 2^n apabila n digit terakhir dari bilangan tersebut habis dibagi 2^n
Contoh :
134576 habis dibagi 8 = 2^3, sebab 576 habis dibagi 8 (576 : 8 = 72)

4971328 habis dibagi 16 = 2^4 sebab 1328 habis dibagi 16

b.) Suatu bilangan habis dibagi 5 apabila digit terakhir dari bilangan tersebut adalah 0 atau 5
Contoh : 67585 dan 457830 adalah bilangan-bilangan yang habis dibagi 5.

c.) Suatu bilangan habis dibagi 3 apabila jumlah digit bilangan tersebut habis dibagi 3
Contoh : 356535 habis dibagi 3 sebab 3 + 5 + 6 + 5 + 3 + 5 = 27 dan 27 habis dibagi 3.

d.) Suatu bilangan habis dibagi 9 apabila jumlah digit bilangan tersebut habis dibagi 9
Contoh : 23652 habis dibagi 9 sebab 2 + 3 + 6 + 5 + 2 = 18 dan 18 habis dibagi 9.

e.) Suatu bilangan habis dibagi 11 apabila selisih antara jumlah digit dari bilangan tersebut pada posisi ganjil dengan jumlah digit dari bilangan tersebut pada posisi genap habis dibagi 11

Contoh : 945351 habis dibagi 11 sebab (9 + 5 + 5) - (4 + 3 + 1) = 11 dan 11 habis dibagi 11.
Contoh bilangan lain yang habis dibagi 11 adalah 53713 dan 245784. 

2.) Jika suatu bilangan habis dibagi a dan juga habis dibagi b, maka bilangan tersebut akan habis dibagi ab dengan syarat a dan b relatif prima
Berlaku sebaliknya.
Contoh : 36 habis dibagi 4 dan 3, maka 36 akan habis dibagi 12.

3.) Misalkan N jika dibagi p akan bersisa r.
Dalam bentuk persamaan N = pq + r dengan p menyatakan pembagi, q menyatakan hasil bagi dan r menyatakan sisa
Persamaan di atas sering pula ditulis N=r (mod p)
----> ga penting

4.) Kuadrat suatu bilangan bulat bulat, habis dibagi 4 atau bersisa 1 jika dibagi 4.
Maka suatu bilangan bulat yang bersisa 2 atau 3 jika dibagi 4, bukanlah bilangan kuadrat.
contoh: 81 dibagi 4 sisa 1, 100 bisa dibagi 4, 121 dibagi 4 sisa 1. Jadi kalo ada bilangan yang jika dibagi 4 sisanya bukan 1 atau 0, maka bilangan tsb bukan bilangan kuadrat

5.) Angka satuan dari bilangan kuadrat adalah 0, 1, 4, 5, 6, 9 (mungkin aja keluar di seri angka)

6.) Bilangan pangkat tiga (kubik) jika dibagi 7 akan bersisa 0, 1 atau 6.

7.) Dua bilangan dikatakan prima relatif, jika faktor persekutuan terbesarnya (FPB) sama dengan 1.
Contoh : 26 dan 47 adalah prima relatif sebab FPB 26 dan 47 ditulis FPB(26,47) = 1

Let's look at some examples in which we test the divisibility of a single whole number.

Example 1:Determine whether 150 is divisible by 2, 3, 4, 5, 6, 9 and 10.
150 is divisible by 2 since the last digit is 0.
150 is divisible by 3 since the sum of the digits is 6 (1+5+0 = 6), and 6 is divisible by 3.
150 is not divisible by 4 since 50 is not divisible by 4.
150 is divisible by 5 since the last digit is 0.
150 is divisible by 6 since it is divisible by 2 AND by 3.
150 is not divisible by 9 since the sum of the digits is 6, and 6 is not divisible by 9.
150 is divisible by 10 since the last digit is 0.
Solution: 150 is divisible by 2, 3, 5, 6, and 10.

Example 2:Determine whether 225 is divisible by 2, 3, 4, 5, 6, 9 and 10.
225 is not divisible by 2 since the last digit is not 0, 2, 4, 6 or 8.
225 is divisible by 3 since the sum of the digits is 9, and 9 is divisible by 3.
225 is not divisible by 4 since 25 is not divisible by 4.
225 is divisible by 5 since the last digit is 5.
225 is not divisible by 6 since it is not divisible by both 2 and 3.
225 is divisible by 9 since the sum of the digits is 9, and 9 is divisible by 9.
225 is not divisible by 10 since the last digit is not 0.
Solution: 225 is divisible by 3, 5 and 9.

Example 3:Determine whether 7,168 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
7,168 is divisible by 2 since the last digit is 8.
7,168 is not divisible by 3 since the sum of the digits is 22, and 22 is not divisible by 3.
7,168 is divisible by 4 since 168 is divisible by 4.
7,168 is not divisible by 5 since the last digit is not 0 or 5.
7,168 is not divisible by 6 since it is not divisible by both 2 and 3.
7,168 is divisible by 8 since the last 3 digits are 168, and 168 is divisible by 8.
7,168 is not divisible by 9 since the sum of the digits is 22, and 22 is not divisible by 9.
7,168 is not divisible by 10 since the last digit is not 0 or 5.
Solution: 7,168 is divisible by 2, 4 and 8.

Example 4:Determine whether 9,042 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
9,042 is divisible by 2 since the last digit is 2.
9,042 is divisible by 3 since the sum of the digits is 15, and 15 is divisible by 3.
9,042 is not divisible by 4 since 42 is not divisible by 4.
9,042 is not divisible by 5 since the last digit is not 0 or 5.
9,042 is divisible by 6 since it is divisible by both 2 and 3.
9,042 is not divisible by 8 since the last 3 digits are 042, and 42 is not divisible by 8.
9,042 is not divisible by 9 since the sum of the digits is 15, and 15 is not divisible by 9.
9,042 is not divisible by 10 since the last digit is not 0 or 5.
Solution:9,042 is divisible by 2, 3 and 6. [IMAGE]

Example 5:Determine whether 35,120 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
35,120 is divisible by 2 since the last digit is 0.
35,120 is not divisible by 3 since the sum of the digits is 11, and 11 is not divisible by 3.
35,120 is divisible by 4 since 20 is divisible by 4.
35,120 is divisible by 5 since the last digit is 0.
35,120 is not divisible by 6 since it is not divisible by both 2 and 3.
35,120 is divisible by 8 since the last 3 digits are 120, and 120 is divisible by 8.
35,120 is not divisible by 9 since the sum of the digits is 11, and 11 is not divisible by 9.
35,120 is divisible by 10 since the last digit is 0.
Solution:35,120 is divisible by 2, 4, 5, 8 and 10. [IMAGE]

Example 6:Is the number 91 prime or composite? Use divisibility when possible to find your answer.
91 is not divisible by 2 since the last digit is not 0, 2, 4, 6 or 8.
91 is not divisible by 3 since the sum of the digits (9+1=10) is not divisible by 3.
91 is not evenly divisible by 4 (remainder is 3).
91 is not divisible by 5 since the last digit is not 0 or 5.
91 is not divisible by 6 since it is not divisible by both 2 and 3.
91 divided by 7 is 13.
Solution:The number 91 is divisible by 1, 7, 13 and 91. Therefore 91 is composite since it has more than two factors.

Summary: Divisibility tests can be used to find factors of large whole numbers quickly, and thus determine if they are prime or composite. When working with large whole numbers, tests for divisibility are more efficient than the traditional factoring method.

gimana?? hehe
kalo ada yang ditanyakan silakan di post
tapi ga njamin bisa njawab :p