A topic in Math 403 in masteral Classes.

1. SLIDESMANIA
Presenter: Sheena Rose P. Tagua
DIOPHANTINE
EQUATIONS

2. SLIDESMANIA
DIOPHANTIN
E
EQUATIONS

3. SLIDESMANIA
Diophantine Equations
● A Diophantine Equation is a
polynomial equation that contains two or
more unknowns. Only integer solution
is acceptable by the Diophantine
equation.
● It is introduced by a mathematician
Diophantus of Alexandria.

4. SLIDESMANIA
Diophantine Equation falls into three
classes:
 Those with no solutions
 Those with only finitely many solutions
 Those with infinitely many solutions

5. SLIDESMANIA
SOLVING
DIOPHANTINE
EQUATIONS
USING
CONGRUENCE

6. SLIDESMANIA
Congruence Method
Congruence Method provide a useful tool in determining the
number of solutions to a Diophantine Equation. It is applied to the
simplest Diophantine Equation, 𝒂𝒙 + 𝒃𝒚 = 𝒄, where a, b, and c
are nonzero integers. These method show that the solution has
either no solutions or infinitely many.

7. SLIDESMANIA
Where a, b, c ∊ Z
Goal: Find integer pairs (x, y)
that satisfy the equation.
𝒂𝒙 + 𝒃𝒚 = 𝒄
Existence of a solution:
Let 𝑑 = 𝑔𝑐𝑑 (𝑎, 𝑏)
𝒂𝒙 + 𝒃𝒚 = 𝒄
𝒉𝒂𝒔 𝒂 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏 𝒊𝒇 𝒂𝒏𝒅
𝒐𝒏𝒍𝒚 𝒊𝒇 𝒅 𝒅𝒊𝒗𝒊𝒅𝒆𝒔 𝒄.

8. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
Example 1:
Find the general
solution of the equation
6𝒙 + 𝟗𝒚 = 𝟏𝟖
6𝒙 + 𝟗𝒚 = 𝟏𝟖
𝑎 = 6 𝑏 = 9 𝑐 = 18
𝑑 = gcd( 6, 9)
𝑑 = 3
3 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 18
∴ 𝒊𝒕 𝒉𝒂𝒔 𝒂 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏
Euclidean Algorithm
9 = 6 1 + 3
6 = 𝟑 2 + 0

9. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
2𝒙 + 𝟑𝒚 = 𝟔
Solution:
6𝒙 + 𝟗𝒚 = 𝟏𝟖
𝟐𝒙 ≡ 𝟔(𝒎𝒐𝒅 𝟑)
𝒙 ≡ 𝟑(𝒎𝒐𝒅 𝟑)
𝒙 = 𝟑 + 𝟑𝒌, 𝑘 ∊ 𝑍
Divides both sides of the equation by the gcd of 3
Convert the equation into congruence in the form of 𝐚𝐱 ≡ 𝒄 (𝒎𝒐𝒅 𝒃)
Divide both sides by 2
Congruence and equality, 𝒂 = 𝒃 + 𝒏𝒌 for some k integers.
General solution for x.

10. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
𝟔(𝟑 + 𝟑𝒌) + 𝟗𝒚 = 𝟏𝟖
Solution: 𝒙 = 𝟑 + 𝟑𝒌, 𝑘 ∊ 𝑍
6𝒙 + 𝟗𝒚 = 𝟏𝟖
𝟗𝒚 = 𝟏𝟖 − 𝟏𝟖 − 𝟏𝟖𝒌
𝟗𝒚 = −𝟏𝟖𝒌
y= −𝟐𝒌, 𝑘 ∊ 𝑍
Substitute 𝐱 = 𝟑 + 𝟑𝒌 to the original equation.
𝟏𝟖 + 𝟏𝟖𝒌 + 𝟗𝒚 = 𝟏𝟖
Solve for y.
Solve.
Divide both side by 9.
General solution for y.

11. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
Example 1:
Find the general
solution of the equation
6𝒙 + 𝟗𝒚 = 𝟏𝟖
Answer:
The general solution are
𝒚 = −𝟐𝒌, 𝑘 ∊ 𝑍
𝒙 = 𝟑 + 𝟑𝒌, 𝑘 ∊ 𝑍

12. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
If the general solution
𝒙 = 𝟑 + 𝟑𝒌, 𝑘 ∊ 𝑍
𝒚 = −𝟐𝒌, 𝑘 ∊ 𝑍
Checking:
Will satisfy the equation
6𝒙 + 𝟗𝒚 = 𝟏𝟖
If 𝑘 = 0
𝑥 = 3 + 3 0 ; 𝑦 = −2 0
𝒙 = 𝟑 𝒚 = 𝟎
𝟔𝒙 + 𝟗𝒚 = 𝟏𝟖
𝟔(𝟑) + 𝟗(𝟎) = 𝟏𝟖
𝟏𝟖 + 𝟎 = 𝟏𝟖
𝟏𝟖 = 𝟏𝟖

13. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
If the general solution
𝒙 = 𝟑 + 𝟑𝒌, 𝑘 ∊ 𝑍
𝒚 = −𝟐𝒌, 𝑘 ∊ 𝑍
Checking:
Will satisfy the equation
6𝒙 + 𝟗𝒚 = 𝟏𝟖
If 𝑘 = 1
𝑥 = 3 + 3 1 ; 𝑦 = −2 1
𝒙 = 𝟔 𝒚 = −𝟐
𝟔𝒙 + 𝟗𝒚 = 𝟏𝟖
𝟔(𝟔) + 𝟗(−𝟐) = 𝟏𝟖
𝟑𝟔 − 𝟏𝟖 = 𝟏𝟖
𝟏𝟖 = 𝟏𝟖

14. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
Example 2:
Using congruence,
solve the linear
Diophantine Equation
48𝒙 + 𝟖𝟒𝒚 = 𝟏𝟒𝟒
48𝒙 + 𝟖𝟒𝒚 = 𝟏𝟒𝟒
𝑎 = 48 𝑏 = 84 𝑐 = 144
𝑑 = gcd( 48, 84)
𝑑 = 12
12 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 144
∴ 𝒊𝒕 𝒉𝒂𝒔 𝒂 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏
Euclidean Algorithm
84 = 48 1 + 36
48 = 36 1 + 12
36 = 12 3 + 0

15. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
4𝒙 + 𝟕𝒚 = 𝟏𝟐
Solution:
48𝒙 + 𝟖𝟒𝒚 = 𝟏𝟒𝟒
𝟒𝒙 ≡ 𝟏𝟐(𝒎𝒐𝒅 𝟕)
𝒙 ≡ 𝟑(𝒎𝒐𝒅 𝟕)
𝒙 = 𝟑 + 𝟕𝒌, 𝑘 ∊ 𝑍
Divides both sides of the equation by the gcd of 12
Convert the equation into congruence in the form of 𝐚𝐱 ≡ 𝒄 (𝒎𝒐𝒅 𝒃)
Divide both sides by 4
Congruence and equality, 𝒂 = 𝒃 + 𝒏𝒌 for some k integers.
General solution for x.

16. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
𝟒𝟖(𝟑 + 𝟕𝒌) + 𝟖𝟒𝒚 = 𝟏𝟒𝟒
Solution: 𝒙 = 𝟑 + 𝟕𝒌, 𝑘 ∊ 𝑍
48𝒙 + 𝟖𝟒𝒚 = 𝟏𝟒𝟒
𝟖𝟒𝒚 = 𝟏𝟒𝟒 − 𝟏𝟒𝟒 − 𝟑𝟑𝟔𝒌
𝟖𝟒𝒚 = −𝟑𝟑𝟔𝒌
𝒚 = −𝟒𝒌, 𝑘 ∊ 𝑍
Substitute 𝐱 = 𝟑 + 𝟕𝒌 to the original equation.
Solve
Solve for y and simplify.
.
Divide both side by 84.
General solution for y.
𝟏𝟒𝟒 + 𝟑𝟑𝟔𝒌 + 𝟖𝟒𝒚 = 𝟏𝟒𝟒

17. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
Example 2:
Find the general
solution of the equation
48𝒙 + 𝟖𝟒𝒚 = 𝟏𝟒𝟒
Answer:
The general solution are
y= −𝟒𝒌, 𝑘 ∊ 𝑍
x= 𝟑 + 𝟕𝒌, 𝑘 ∊ 𝑍

18. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
If the general solution
𝒙 = 𝟑 + 𝟕𝒌, 𝑘 ∊ 𝑍
𝒚 = −𝟒𝒌, 𝑘 ∊ 𝑍
Checking:
Will satisfy the equation
𝟒𝟖𝒙 + 𝟖𝟒𝒚 = 𝟏𝟒𝟒
If 𝑘 = −3
𝑥 = 3 + 7 −3 ; 𝑦 = −4 −3
𝒙 = −𝟏8 𝒚 = 𝟏𝟐
𝟒𝟖𝒙 + 𝟖𝟒𝒚 = 𝟏𝟒𝟒
𝟒𝟖(−𝟏𝟖) + 𝟖𝟒(𝟏𝟐) = 𝟏𝟐𝟑
-8𝟔𝟒 + 𝟏𝟎𝟎𝟖 = 𝟏𝟒𝟒
𝟏𝟒𝟒 = 𝟏𝟒𝟒

19. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
Example 3:
Using the congruence,
solve the linear
Diophantine equation
𝟗𝒙 + 𝟔𝒚 = 𝟐𝟓
𝟗𝒙 + 𝟔𝒚 = 𝟐𝟓
𝑎 = 9 𝑏 = 6 𝑐 = 25
𝑑 = gcd( 9, 6)
𝑑 = 3
3 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 25
Euclidean Algorithm
9 = 6 1 + 3
6 = 𝟑 2 + 0

20. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
Example 3:
Using the congruence,
solve the linear
Diophantine equation
𝟗𝒙 + 𝟔𝒚 = 𝟐𝟓
Answer:
I𝒕 𝒉𝒂𝒔 𝒏𝒐 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏

21. SLIDESMANIA
SHORT
ACTIVITY

22. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
Problem 1:
Find the general
solution of the equation
𝟓𝟎𝒙 + 𝟔𝟎𝒚 = 𝟑𝟎𝟎
𝑑 = gcd( 50, 60)
𝑑 = 10
10 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 300
Euclidean Algorithm
60 = 50 1 + 10
50 = 10 5 + 0
∴ I𝒕 𝒉𝒂𝒔 𝒂 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏
𝟓𝟎𝒙 + 𝟔𝟎𝒚 = 𝟑𝟎𝟎
𝑎 = 50 𝑏 = 60 𝑐 = 300

23. SLIDESMANIA
― Irene M. Pepperberg
𝟓𝟎𝒙 + 𝟔𝟎𝒚 = 𝟑𝟎𝟎
𝟓𝒙 + 𝟔𝒚 = 𝟑𝟎
𝟓𝒙 ≡ 𝟑𝟎 (𝒎𝒐𝒅 𝟔)
Divides both sides of the equation by the gcd of 10
Convert the equation into congruence in the form of ax≡ 𝒄 (𝒎𝒐𝒅 𝒃)
𝟓𝒙 ≡ 𝟑𝟎 (𝒎𝒐𝒅 𝟔) Divide both sides by 5
𝒙 ≡ 𝟔 (𝒎𝒐𝒅 𝟔)
𝒙 = 𝟔 + 𝟔𝒌, 𝑘 ∊ 𝑍
Congruence and equality, 𝒂 = 𝒃 + 𝒏𝒌 for some k integers.
General solution for y.

24. SLIDESMANIA
― Irene M. Pepperberg
𝟓𝟎(𝟔 + 𝟔𝒌) + 𝟔𝟎𝒚 = 𝟑𝟎𝟎
𝟑𝟎𝟎 + 𝟑𝟎𝟎𝒌 + 𝟔𝟎𝒚 = 𝟑𝟎𝟎
𝟔𝟎𝒚 = 𝟑𝟎𝟎 − 𝟑𝟎𝟎 − 𝟑𝟎𝟎𝒌
𝟔𝟎𝒚 = −𝟑𝟎𝟎𝒌
𝒚 = −𝟓𝒌, 𝑘 ∊ 𝑍
𝒙 ≡ 𝟔 + 𝟔𝒌, 𝑘 ∊ 𝑍 Substitute 𝐱 = 𝟔 + 𝟔𝒌 to the original equation.
Solve
Solve for y.
Simplify.
Divide both side by 60.
General solution for x.

25. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
Problem 1:
Find the general
solution of the equation
𝟓𝟎𝒙 + 𝟔𝟎𝒚 = 𝟑𝟎𝟎
Answer
𝒙 = 𝟔 + 𝟔𝒌, 𝑘 ∊ 𝑍
𝒚 = −𝟓𝒌, 𝑘 ∊ 𝑍

26. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
If the general solution
Checking:
Will satisfy the equation
𝟓𝟎𝒙 + 𝟔𝟎𝒚 = 𝟑𝟎𝟎
If 𝑘 = 1
𝑥 = 6 + 6 1 ; 𝑦 = −5 1
𝒙 = 𝟏𝟐 𝒚 = −𝟓
50𝒙 + 𝟔𝟎𝒚 = 𝟑𝟎𝟎
𝟓𝟎(𝟏𝟐) + 𝟔𝟎(−𝟓) = 𝟑𝟎𝟎
𝟔𝟎𝟎 − 𝟑𝟎𝟎 = 𝟑𝟎𝟎
𝟑𝟎𝟎 = 𝟑𝟎𝟎
𝒙 = 𝟔 + 𝟔𝒌, 𝑘 ∊ 𝑍
𝒚 = −𝟓𝒌, 𝑘 ∊ 𝑍

27. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
Problem 2:
Find the general
solution of the equation
2𝒙 + 𝟒𝒚 = 𝟐𝟏
2𝒙 + 𝟒𝒚 = 𝟐𝟏
𝑎 = 2 𝑏 = 4 𝑐 = 21
𝑑 = gcd( 2, 4)
𝑑 = 2
2 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 21
Euclidean Algorithm
4 = 2 2 + 0

28. SLIDESMANIA
Solving Diophantine Equation Using
Congruence
Problem 2:
Find the general
solution of the equation
2𝒙 + 𝟒𝒚 = 𝟐𝟏
Answer
I𝒕 𝒉𝒂𝒔 𝒏𝒐 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏

29. SLIDESMANIA
PYTHAGOREAN
TRIPLES

30. SLIDESMANIA
Pythagorean Triples
● Pythagorean triples are three positive integers
which satisfy the Pythagorean theorem.
• Pythagorean theorem states that in any right -
angle triangle, the square of the hypotenuse is
equal to the sum of the squares of the other two
sides of the right triangle.

31. SLIDESMANIA
Definition of a Right Triangle – a triangle with exactly
one right angle.
The hypotenuse is always across from the right angle
and is always the longest side.
hypotenuse
The legs are always the two sides that form the right
angle and are shorter than the hypotenuse.
legs

32. SLIDESMANIA
Pythagorean Theorem – In a right triangle, the
square of the hypotenuse is equal to the sum of
the squares of the two legs.
a
b
c
c2 = a2 + b2

33. SLIDESMANIA
Pythagorean Triples
● Three integers that make the equation
c2 = a2 + b2 true are called Pythagorean
Triples.

34. SLIDESMANIA
How to form Pythagorean
Triples?
Case 1: If the number is odd:
If x is odd, then the Pythagorean triples 𝑥,
𝑥2
2
− 0.5 ;
𝑥2
2
+ 0.5

35. SLIDESMANIA
How to form Pythagorean
Triples?
Consider an example (7, 24, 25)
𝑥 = 7, 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑜𝑑𝑑
𝑥2
2
− 0.5 =
72
2
− 0.5 =
49
2
− 0.5 = 24.5 − 0.5 = 𝟐𝟒
𝑥2
2
+ 0.5 =
72
2
+ 0.5 =
49
2
+ 0.5 = 24.5 + 0.5 = 𝟐5

36. SLIDESMANIA
Consider that the first number in
Pythagorean triples is 9.
𝑥 = 9, 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑜𝑑𝑑
𝑥2
2
− 0.5 =
92
2
− 0.5 =
81
2
− 0.5 = 40.5 − 0.5 = 𝟒𝟎
𝑥2
2
+ 0.5 =
92
2
+ 0.5 =
81
2
+ 0.5 = 40.5 + 0.5 = 𝟒𝟏
9, 40, 41

37. SLIDESMANIA
Consider that the first number in
Pythagorean triples is 3.
𝑥 = 3, 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑜𝑑𝑑
𝑥2
2
− 0.5 =
32
2
− 0.5 =
9
2
− 0.5 = 4.5 − 0.5 = 𝟒
𝑥2
2
+ 0.5 =
32
2
+ 0.5 =
9
2
+ 0.5 = 4.5 + 0.5 = 𝟓
3, 4, 5

38. SLIDESMANIA
How to form Pythagorean
Triples?
Case 2: If the number is even:
If x is even, then the Pythagorean triple 𝑥,
𝑥
2
2
− 1 ;
𝑥
2
2
−1

39. SLIDESMANIA
How to form Pythagorean
Triples?
Consider an example (16, 63, 65)
𝑥 = 16, 𝑖𝑠 𝑒𝑣𝑒𝑛
𝑥
2
2
− 1 =
16
2
2
− 1 = 8 2 − 1 = 64 − 1 = 𝟔𝟑
𝑥
2
2
+ 1 =
16
2
2
+ 1 = 8 2 + 1 = 64 + 1 = 𝟔𝟓

40. SLIDESMANIA
Consider that the first number in
Pythagorean triples is 18.
𝑥 = 18, 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑒𝑣𝑒𝑛
𝑥
2
2
− 1 =
18
2
2
− 1 = 9 2 − 1 = 81 − 1 = 𝟖𝟎
𝑥
2
2
+ 1 =
18
2
2
+ 1 = 9 2
+ 1 = 81 + 1 = 𝟖𝟐
18, 80, 82

41. SLIDESMANIA
Prove that (18, 80, 81) is a Pythagorean triple.
Solution: a= 18, b= 80, c= 81
𝒂𝟐
+ 𝒃𝟐
= 𝒄𝟐
𝟏𝟖𝟐 + 𝟖𝟎𝟐 = 𝟖𝟐𝟐
𝟑𝟐𝟒 + 𝟔𝟒𝟎𝟎 = 𝟔𝟕𝟐𝟒
𝟔𝟕𝟐𝟒 = 𝟔𝟕𝟐𝟒
Hence, the given set of
integers satisfies the
Pythagorean Theorem,
(5, 12, 13) is a
Pythagorean triples.

42. SLIDESMANIA
Prove that (8, 15, 17) is a Pythagorean triple.
Solution: a= 8, b= 15, c= 17
𝒂𝟐
+ 𝒃𝟐
= 𝒄𝟐
𝟖𝟐 + 𝟏𝟓𝟐 = 𝟏𝟕𝟐
𝟔𝟒 + 𝟐𝟐𝟓 = 𝟐𝟖𝟗
𝟐𝟖𝟗 = 𝟐𝟖𝟗
Hence, the given set of
integers satisfies the
Pythagorean Theorem,
(8, 15, 17) is a
Pythagorean triples.

43. SLIDESMANIA
6
8
c
𝒂𝟐
+ 𝒃𝟐
= 𝒄𝟐
𝒂 = 𝟔 𝒃 = 𝟖 𝒄 =?
𝟔𝟐
+ 𝟖𝟐
= 𝒄𝟐
𝟑𝟔 + 𝟔𝟒 = 𝒄𝟐
𝟏𝟎𝟎 = 𝒄𝟐
𝟏𝟎𝟎 = 𝒄𝟐
𝟏𝟎 = 𝒄
(6, 8, 10)

44. SLIDESMANIA
6 𝟑
b
12
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
𝒂 = 𝟔 𝟑 𝒃 =? 𝒄 = 𝟏𝟐
(𝟔 𝟑)
𝟐
+𝒃𝟐 = 𝟏𝟐𝟐
𝟑𝟔 𝟗 + 𝒃𝟐 = 𝟏𝟒𝟒
𝟑𝟔(𝟑) + 𝒃𝟐 = 𝟏𝟒𝟒
𝟏𝟎𝟖 + 𝒃𝟐 = 𝟏𝟒𝟒
𝒃𝟐
= 𝟏𝟒𝟒 − 𝟏𝟎𝟖
𝒃𝟐 = 𝟑𝟔
𝒃 = 𝟔
(𝟔 𝟑, 𝟔, 𝟏𝟐)

45. SLIDESMANIA
5 𝟑
5
c
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
𝒂 = 𝟓 𝒃 = 𝟓 𝟑 𝒄 =?
(𝟓𝟐 + (𝟓 𝟑)
𝟐
= 𝒄𝟐
𝟐𝟓 + 𝟐𝟓 𝟗 = 𝒄𝟐
𝟐𝟓 + 𝟐𝟓 𝟑 = 𝒄𝟐
𝟐𝟓 + 𝟕𝟓 = 𝒄𝟐
𝟏𝟎𝟎 = 𝒄𝟐
𝟏𝟎𝟎 = 𝒄𝟐
𝟏𝟎 = 𝒄

46. SLIDESMANIA
9
a
15
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
𝒂 =? 𝒃 = 𝟗 𝒄 = 𝟏𝟓
(𝒂𝟐 + 𝟗𝟐= 𝟏𝟓𝟐
𝒂𝟐
+ 𝟖𝟏 = 𝟐𝟐𝟓
𝒂𝟐 = 𝟐𝟐𝟓 − 𝟖𝟏
𝒂𝟐 = 𝟏𝟒𝟒
𝒂𝟐 = 𝟏𝟒𝟒
𝒂 = 𝟏𝟐

47. SLIDESMANIA
24
a
25
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
𝒂 =? 𝒃 = 𝟐𝟒 𝒄 = 𝟐𝟓
(𝒂𝟐 + 𝟐𝟒𝟐= 𝟐𝟓𝟐
𝒂𝟐
+ 𝟓𝟕𝟔 = 𝟔𝟐𝟓
𝒂𝟐 = 𝟔𝟐𝟓 − 𝟓𝟕𝟓
𝒂𝟐 = 𝟓𝟎
𝒂𝟐 = 𝟓𝟎
𝒂 = 𝟓 𝟐

48. SLIDESMANIA
Thank you for
listening!