def sol_count(a, b, m):
"""
Returns number of solutions of ax ≡ b (mod m) that are least residue (mod m)
Parameters
----------
a : int
denotes a in ax ≡ b (mod m)
b : int
denotes b in ax ≡ b (mod m)
m : int
denotes m in ax ≡ b (mod m)
"""
if (isSol(a, b, m) == False):
return 0
elif (isCoprime(a, m)):
return 1
else:
return gcd(a, m)
def fermat_prime_checker(p, a):
"""
Checks if p is prime using fermat's principle
If p is prime and gcd(a, p) = 1, then a^(p-1) ≡ 1 (mod p)
Parameters
----------
p : int
denotes a natural number that has to be checked for prime
a : int
denotes a natural number co-prime with p
return : bool
return true if p is prime otherwise false
"""
if (isCoprime(a, p) == False):
raise ValueError("a must be co-prime with p")
if (p == 1):
return False
return (math.pow(a, p - 1) % p) == (1 % p)
def order(a, m):
"""
Returns order of a modulo m
order of a modulo m is the smallest positive integer t such that a^t ≡ 1 (mod m)
Parameters
----------
a : int
denotes positive integer a in a^t ≡ 1 (mod m)
m : int
denotes positive integer m in a^t ≡ 1 (mod m)
return : int
returns least integet t such that a^t ≡ 1 (mod m)
"""
if (isCoprime(a, m) == False):
raise ValueError("a and m must be co-prime")
i = 1
while True:
if (((math.pow(a, i)) % m) == 1):
break
i = i + 1
return i
def test_isCoprime2():
a = 100
b = 77895345
assert isCoprime(a, b) == False
def test_isCoprime1():
a = 343
b = 10000
assert isCoprime(a, b) == True