#### Diophantine Equation

```from __future__ import annotations

def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
"""
Diophantine Equation : Given integers a,b,c ( at least one of a and b != 0), the
diophantine equation a*x + b*y = c has a solution (where x and y are integers)
iff gcd(a,b) divides c.

GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )

>>> diophantine(10,6,14)
(-7.0, 14.0)

>>> diophantine(391,299,-69)
(9.0, -12.0)

But above equation has one more solution i.e., x = -4, y = 5.
That's why we need diophantine all solution function.

"""

assert (
c % greatest_common_divisor(a, b) == 0
)  # greatest_common_divisor(a,b) function implemented below
(d, x, y) = extended_gcd(a, b)  # extended_gcd(a,b) function implemented below
r = c / d
return (r * x, r * y)

def diophantine_all_soln(a: int, b: int, c: int, n: int = 2) -> None:
"""
Lemma : if n|ab and gcd(a,n) = 1, then n|b.

Finding All solutions of Diophantine Equations:

Theorem : Let gcd(a,b) = d, a = d*p, b = d*q. If (x0,y0) is a solution of
Diophantine Equation a*x + b*y = c.  a*x0 + b*y0 = c, then all the
solutions have the form a(x0 + t*q) + b(y0 - t*p) = c,
where t is an arbitrary integer.

n is the number of solution you want, n = 2 by default

>>> diophantine_all_soln(10, 6, 14)
-7.0 14.0
-4.0 9.0

>>> diophantine_all_soln(10, 6, 14, 4)
-7.0 14.0
-4.0 9.0
-1.0 4.0
2.0 -1.0

>>> diophantine_all_soln(391, 299, -69, n = 4)
9.0 -12.0
22.0 -29.0
35.0 -46.0
48.0 -63.0

"""
(x0, y0) = diophantine(a, b, c)  # Initial value
d = greatest_common_divisor(a, b)
p = a // d
q = b // d

for i in range(n):
x = x0 + i * q
y = y0 - i * p
print(x, y)

def greatest_common_divisor(a: int, b: int) -> int:
"""
Euclid's Lemma :  d divides a and b, if and only if d divides a-b and b

Euclid's Algorithm

>>> greatest_common_divisor(7,5)
1

Note : In number theory, two integers a and b are said to be relatively prime,
mutually prime, or co-prime if the only positive integer (factor) that
divides both of them is 1  i.e., gcd(a,b) = 1.

>>> greatest_common_divisor(121, 11)
11

"""
if a < b:
a, b = b, a

while a % b != 0:
a, b = b, a % b

return b

def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
"""
Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers
x and y, then d = gcd(a,b)

>>> extended_gcd(10, 6)
(2, -1, 2)

>>> extended_gcd(7, 5)
(1, -2, 3)

"""
assert a >= 0 and b >= 0

if b == 0:
d, x, y = a, 1, 0
else:
(d, p, q) = extended_gcd(b, a % b)
x = q
y = p - q * (a // b)

assert a % d == 0 and b % d == 0
assert d == a * x + b * y

return (d, x, y)

if __name__ == "__main__":
from doctest import testmod

testmod(name="diophantine", verbose=True)
testmod(name="diophantine_all_soln", verbose=True)
testmod(name="extended_gcd", verbose=True)
testmod(name="greatest_common_divisor", verbose=True)
```  