Calculations with congruences | Patterns of Thought plus AI Reasoning

congruence_addition.py

def congruence_addition(a, a_prime, b, b_prime, m):
    """Proves congruence addition: a' + b' ≡ a + b (mod m)"""
    if a_prime % m == a % m and b_prime % m == b % m:
        result_prime = (a_prime + b_prime) % m
        result = (a + b) % m
        if result_prime == result:
            print(f"{a_prime} + {b_prime} ≡ {a} + {b} (mod {m}) is TRUE")
            return True
        else:
            print(f"{a_prime} + {b_prime} ≡ {a} + {b} (mod {m}) is FALSE")
            return False
    else:
        print(f"Conditions not met: a' ≡ a (mod {m}) and b' ≡ b (mod {m}) must hold")
        return False


def congruence_subtraction(a, a_prime, b, b_prime, m):
    """Proves congruence subtraction: a' - b' ≡ a - b (mod m)"""
    if a_prime % m == a % m and b_prime % m == b % m:
        result_prime = (a_prime - b_prime) % m
        result = (a - b) % m
        if result_prime == result:
            print(f"{a_prime} - {b_prime} ≡ {a} - {b} (mod {m}) is TRUE")
            return True
        else:
            print(f"{a_prime} - {b_prime} ≡ {a} - {b} (mod {m}) is FALSE")
            return False
    else:
        print(f"Conditions not met: a' ≡ a (mod {m}) and b' ≡ b (mod {m}) must hold")
        return False


def congruence_multiplication(a, a_prime, b, b_prime, m):
    """Proves congruence multiplication: a' * b' ≡ a * b (mod m)"""
    if a_prime % m == a % m and b_prime % m == b % m:
        result_prime = (a_prime * b_prime) % m
        result = (a * b) % m
        if result_prime == result:
            print(f"{a_prime} * {b_prime} ≡ {a} * {b} (mod {m}) is TRUE")
            return True
        else:
            print(f"{a_prime} * {b_prime} ≡ {a} * {b} (mod {m}) is FALSE")
            return False
    else:
        print(f"Conditions not met: a' ≡ a (mod {m}) and b' ≡ b (mod {m}) must hold")
        return False



# Example usage (using the same values from the previous response):
m = 5
a = 7
a_prime = 12
b = 3
b_prime = 8

congruence_addition(a, a_prime, b, b_prime, m)       # Output: 20 ≡ 10 (mod 5) is TRUE
congruence_subtraction(a, a_prime, b, b_prime, m)    # Output: 4 ≡ 4 (mod 5) is TRUE
congruence_multiplication(a, a_prime, b, b_prime, m) # Output: 96 ≡ 21 (mod 5) is TRUE

# Example with values that don't satisfy the congruence:
a_prime = 13 # Now a' is NOT congruent to a (mod 5)
congruence_addition(a, a_prime, b, b_prime, m)  # Output: Conditions not met...


# Another example showing negative results and how they are handled.
a = 2
a_prime = 7
b = 5
b_prime = 10
m= 7

congruence_subtraction(a, a_prime, b, b_prime, m) # Output: -3 ≡ -3 (mod 7) is TRUE.  (-3 mod 7 is 4)