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Divisibility: Ab Ab Are All Ab Ba

1) The document discusses divisibility, which refers to whether one integer is divisible by another. It defines what it means for an integer m to be divisible by a nonzero integer n. 2) It then provides some basic properties of divisibility and several divisibility rules to determine if a number is divisible by integers from 2 to 13 based on patterns in the digits. 3) The document concludes with examples of divisibility problems to solve involving determining remainders when dividing integers and identifying missing digits in numbers based on their divisibility.

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193 views2 pages

Divisibility: Ab Ab Are All Ab Ba

1) The document discusses divisibility, which refers to whether one integer is divisible by another. It defines what it means for an integer m to be divisible by a nonzero integer n. 2) It then provides some basic properties of divisibility and several divisibility rules to determine if a number is divisible by integers from 2 to 13 based on patterns in the digits. 3) The document concludes with examples of divisibility problems to solve involving determining remainders when dividing integers and identifying missing digits in numbers based on their divisibility.

Uploaded by

NAJIB F.F
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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Pertemuan 2

DIVISIBILITY
Back in elementary school, we learned four fundamental operations on numbers (integers),

namely, addition    , subtraction    , multiplication  or  , and division ( or or ) . For

any two integers a and b, their sum a  b, differences a  b and b  a , and product ab are all
a
integers, while their quotients a  b (or a b or ) and b  a not necessarily integers.
b
For an integer m and a nonzero integer n, we say that m is divisible by n or n divides
m
m if there is an integer k such that m  kn; that is, is an integer. We denote this by n m. If m
n
is divisible by n , then m is called a multiple of n; and n is called a divisor (or factor) of m.

Because 0  0  n, it follows that n 0 for all integers n. For a fixed integer n, the multiples

of n are 0,  n, 2n,...... Hence it is not difficult to see that there is a multiple of n among every
n consecutive integer. If m is not divisible by n , then we write n | m. (Note that 0 | m for all
nonzero integers m , since m  0  k  0 for all integers k. )

Proposition 1.1. Let x, y and z be integers. We have the following basic properties:

a) x x (reflexivity property);

b) If x y and y z , then x z (transitivity property);

c) If x y and y  0, then x  y ;

d) If x y and x z , then x  y   z for any integers  and  ;

e) If x y and x y  z , then x z ;

f) If x y and y x , then x  y ;

y
g) If x y and y  0, then y;
x

h) for x y and z  0, x y if and only if xz yz .


Pertemuan 2

Proposition 1.2 Some divisibility rules


A number is divisible by
1) 2 if and only if its last digit is even.
2) 3 if and only if the sum of its digits is divisible by 3.
3) 4 if and only if itstwo last digits from a number divisible by 4.
4) 5 if and only if its last digit is either 0 or 5.
5) 6 if and only if it is divisible by both 2 and 3.
6) 7 if and only if taking the last digit, doubling it, and subtracting the result from the rest of the
number gives the answer which is divisible by 7 (including 0).
7) 8 if and only if its three last digits from a number divisible by 8.
8) 9 if and only if the sum of its digits is divisible by 9.
9) 10 if and only if it ends with 0.
10) 11 if and only if alternately adding and subtracting the digits from left to right, the result
(including 0) is divisible by 11.
11) 12 if and only if it is divisible by both 3 and 4.
12) 13 if and only if deleting the last digit from the number, then subtracting 9 times thedeleted
digit from the remaining number gives the answer which is is divisible by 13.

LATIHAN SOAL

1. Show that 1001 divides12013 + 22013 + 32013 + ⋯ + 10002013


2. Misalkan x dan y bilangan bulat. Buktikan 2 x  3 y habis dibagi 17 jika dan hanya jika 9 x  5 y habis

dibagi 17.

3. Tunjukkan bahwa untuk setiap bilangan bulat positif n maka 121n  25n  1900n  (4)n habis dibagi
2000
4. Diberikan bahwa 34!= 295 232 799 cd9 604 140 847 618 609 643 5ab 000 000. Tentukan digit a, b, c dan

d.
5. Sebuah bilangan 8 digit 123456d3, jika dibagi 6 memberi sisa 5. Tentukan nilai d yang mungkin.


6. Tentukan nilai digit k dalam persamaan 2k 99561  3  523  k  
2

7. Buktikan bahwa 49  36 habis dibagi 13


n n

8. Tentukan semua bilangan asli n agar 2n  1 habis dibagi 3

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