The document discusses divisibility rules and algorithms for integers. It provides divisibility rules for integers 2 through 20, describing how to determine if a number is divisible by each factor. For example, a number is divisible by 2 if the last digit is even. It also describes algorithms for determining the greatest common divisor of two numbers, such as the Euclidean algorithm which uses successive division. The Goldschmidt division algorithm iteratively multiplies factors to converge the divisor to 1. Diophantine equations allow only integer solutions.

1. DIVISIBILITY THEORY OF INTEGERS
• “Divisible by” means when you divide one
number with another number the result
should be whole number with zero remainder.

2. DIVISIBILITY RULES
Divisibility Rule for 2
 All even numbers are divisible by 2
 The last digit must be even number
0, 2, 4, 6, 8.
Example:
456- last digit 6 is an even number.
23-last digit 3 is not an even number

3.  Divisibility Rule for 3
 The sum of digits in given number should
be divisible by 3.
Example :
3789 is divisible by 3 –> sum
3+7+8+9= 27 is divisible by 3.
566 is not divisible by 3 –> sum
4+6+6=17, 17 is not divisible by 3.

4.  Divisibility Rule for 6
 The number must be divisible by 2 and 3.
Because 2 and 3 are prime factors of 6.
Example:
7563 894 is divisible by 6 –> last digit
is 4, so divisible by 2, and sum
7+5+6+3+8+9+4 = 42 is divisible by 3.

5.  Divisibility Rule for 7
 Twice the last digit and subtract it from
remaining number in given number, result
result must be divisible by 7.
Example:
154 is divisible by 7
15-(2*4)=7
7÷7=1
343 is divisible 7 –>
34 – (2*3) = 28
28÷ 7=4

6. Divisibility Rule for 8
 The number formed by last three digits in
given number must be divisible by 8.
Example:
• 234568 is divisible by 8 –> 568 is
divisible by 8.
• 8742 is not divisible by 8 –> 742 is not
divisible by 8.

7.  Divisibility Rule for 9
 Same as 3. Sum of digits in given number
must be divisible by 9.
Example:
• 456786 is divisible by 9
4+5+6+7+8+6 = 36 is divisible by 9.
• 87956 is not divisible by 9
8+7+9+5+6 = 35 is not divisible by 9.

8.  Divisibility Rule for 10
 Last digit must be 0.
Example:
1220-last digit is 0.
2231-last digit is 1.

9.  Divisibility Rule for 11
 Form the alternating sum of digits. The result
must be divisible by 11.
Example:
• 416042 is divisible by 11 –>+ 4-1+6-
0+4-2 = 11, 11 is divisible by 11.
• 8219543574 is divisible by 11
8-2+1-9+5-4+3-5+7-4 = 0 is divisible by
by 11.

10.  Divisibility Rule for 12
 The number must be divisible by 3 and 4.
Because (3* (22
)) are prime factors of 12.
Example:
462157692 is divisible by 12
last 2 digits 92, so divisible by 4, and
sum 4+6+2+1+5+7+6+9+2 = 42 is
divisible by 3.

11.  Divisibility Rule for 13
 Add 4 times the last digit to the remaining
truncated number. Repeat the step as
necessary. If the result is divisible by 13,
the original number is also divisible by 13

12. Example:
3146; 314 + (4*6) = 338;
33 + (4* 8) = 65
Since 65 is divible by 13, the original number
3146 is also divisible

13.  Divisibility 14
 The number must be divisible by 2 and 7.
Because 2 and 7 are prime factors of 14.
Divisibility 15
 The number should be divisible by 3 and 5.
Because 3 and 5 are prime factors of 15.

14.  Divisibility 16
 The number formed by last four digits in given
number must be divisible by 16.
Example:
7852176 is divisible by 16 –> 2176 is
divisible by 16.

15.  Divisibility 17
 Multiply last digit with 5 and subtract it from
remaining number in given number, result
must be divisible by 17.
Example:

16. Example : 765 divisible by 17
1.Multiply the last digit by 5. That is 5 * 5 = 25
2.Subtract it from the rest that is 76 – 25 = 51
and 51 is divisible by (17 * 3 = 51) 51/ 3 so
765 is divisible by 17

17.  Divisibility 18
 The number should be divisible by 2 and
9. Because (2*(32
)) are prime factors of 18.
Example: 36/2=18
36/9=4

18.  Divisibility 19
 Multiply last digit with 2 and add it to
remaining number in given number, result
must be divisible by 19. (You can again apply
this to check for divisibility by 19.)
Example:
269- (9*2)+39=57
57÷19=3

19.  Divisibility 20

 The number formed by last two digits in
given number must be divisible by 20.
Example:
2480-80 is divisible by 20

20. DIVISION ALGORITHM
A division algorithm is an algorithm which given
two integers N and D, Computers their quotient
and/or remainder, the result of division. Some
are applied by hand, while others are employed
by digital circuit designs and software.

21. Discussions will refer to the form, where:
• N = Numerator (dividend)
• D = Denominator (divisor)
is the output, and
• Q = Quotient
• R = Remainder

22. The Greatest Common Divisor (GCD)
The greatest common divisor (gcd) of two or more integers,
which are not all zero, is the largest positive integer that divides
each of the integers.
The greatest common divisor is also known as the greatest
common factor, highest common factor, greatest common
measure, or highest common divisor.

23. Example 1:
What is the greatest common divisor of 20 and 40?
Another way to say this is.
gcd(20,40) = 20

24. Example 2:
What is the greatest common divisor of 21 and
30?
gcd(21,30) = 3
21 : 1, 3, 7, 21
30 : 1, 2, 3, 5, 6, 10, 15, 30

25. Factorization
105
Λ
Gcd(105,30)
5 21
Λ
3 7
30
Λ
3 10
Λ
2 5
105 : 3 x 5 x 7 30 : 2 x 3 x 5
= 15

26. Goldschmidt – 5 division uses an iterative process of
repeatedly multiplying both the dividend and the
divisor by a common factor F chosen such that the
divisor converges to one.
Steps for Goldschmidt division are:
1.Generate an estimate for the multiplication factor F,
2.Multiply the dividend and divisor by F,
3.If the divisor is sufficiently close to 1, return the
dividend, otherwise look to step 1.
• The Euclidean Algorithm
This method asks you to perform successive
division, first of the smaller of the two numbers
into larger, followed by the resulting remainder
divided into the divisor of each division until
the remainder is equal to zero.(a,b)integer.

27.  The Euclidean algorithm is a way to find the
Greatest Common Divisor of two positive
integers, a and b.
• Formal description of the Euclidean
algorithm:
1. Input
2. Output
3. Internal computation

28. Goldschmidt – 5 division uses an iterative process of
repeatedly multiplying both the dividend and the
divisor by a common factor F chosen such that the
divisor converges to one.
Steps for Goldschmidt division are:
1.Generate an estimate for the multiplication factor F,
2.Multiply the dividend and divisor by F,
3.If the divisor is sufficiently close to 1, return the
dividend, otherwise look to step 1.
Example:
(3084,
1424)
a=3084
b=1424

29. Example: 528, 142
528
142
3
426
102
−−−− −
r 102
142
1
102
102
_________
40
r 40
102
40
2
80
_________
22
r 22

30. 40
22
1
22
________
18
r 18
22
18
1
18
________
6
r 6
18
6
3
18
________
0

31. Goldschmidt – 5 division uses an iterative process of
repeatedly multiplying both the dividend and the
divisor by a common factor F chosen such that the
divisor converges to one.
Steps for Goldschmidt division are:
1.Generate an estimate for the multiplication factor F,
2.Multiply the dividend and divisor by F,
3.If the divisor is sufficiently close to 1, return the
dividend, otherwise look to step 1.
 Diophantine Equation
 is an equation where only integer solution are
accepted. This implies that Diophantine
equation becomes harder (or even
impossible) to solve than equation that do
not have this restriction.
 ax + by = c,

32. Goldschmidt – 5 division uses an iterative process of
repeatedly multiplying both the dividend and the
divisor by a common factor F chosen such that the
divisor converges to one.
Steps for Goldschmidt division are:
1.Generate an estimate for the multiplication factor F,
2.Multiply the dividend and divisor by F,
3.If the divisor is sufficiently close to 1, return the
dividend, otherwise look to step 1.
Ax + b = c,
 a,b,c,x,y antigen be solve by the method of
solution depends of the coefficients a, b, and
c,.

33. Goldschmidt – 5 division uses an iterative process of
repeatedly multiplying both the dividend and the
divisor by a common factor F chosen such that the
divisor converges to one.
Steps for Goldschmidt division are:
1.Generate an estimate for the multiplication factor F,
2.Multiply the dividend and divisor by F,
3.If the divisor is sufficiently close to 1, return the
dividend, otherwise look to step 1.
Thank You…