Modular arithmetic

Understand the idea

Modulo groups integers by equal remainders. Reduce after addition and multiplication to bound intermediate values. Division needs a modular inverse, which exists only when divisor and modulus are coprime.

Work through an example

Modulo 7, 3×5=15≡1, so 5 is the inverse of 3. Modulo 6, 2 has no inverse because gcd(2,6)=2.

Time and space complexity

Modular operations depend on integer size; fast exponentiation uses O(log exponent) multiplications.

Why it works

Reducing equivalent values preserves addition and multiplication modulo m.

Common mistakes

• Ordinary integer division is not modular division.
• Fermat's inverse exponent needs a prime modulus and nonzero residue.

Chapter lab · Python implementation

Arithmetic assumptions affect algorithm correctness. The lab uses Euclid for GCD, modular repeated squaring, and a square-matrix rotation. Python integers avoid fixed-width overflow but grow in storage and arithmetic cost. Treat coordinate conventions and modular inverses explicitly rather than relying on memorized formulas.

This shared lab connects the chapter's topics. Follow the current lesson's worked example first, then trace the complete implementation. Additional algorithms described in topic exercises are variations you can implement independently.

def gcd(a,b):
    a,b=abs(a),abs(b)
    while b: a,b=b,a%b
    return a

def mod_power(base,exponent,modulus):
    if exponent<0 or modulus<1: raise ValueError('invalid exponent/modulus')
    result=1%modulus; base%=modulus
    while exponent:
        if exponent&1: result=result*base%modulus
        base=base*base%modulus; exponent//=2
    return result

def rotate(matrix):
    n=len(matrix)
    if any(len(row)!=n for row in matrix): raise ValueError('square required')
    for r in range(n):
        for c in range(r+1,n):
            matrix[r][c],matrix[c][r]=matrix[c][r],matrix[r][c]
    for row in matrix: row.reverse()

def count_primes(limit):
    if limit<2: return 0
    prime=bytearray([1])*limit
    prime[0]=prime[1]=0
    p=2
    while p*p<limit:
        if prime[p]:
            for multiple in range(p*p,limit,p):
                prime[multiple]=0
        p+=1
    return sum(prime)

def digit_power(base,digits,modulus):
    if modulus<1 or any(not 0<=d<=9 for d in digits):
        raise ValueError('positive modulus and decimal digits required')
    result=1%modulus
    for digit in digits:
        result=mod_power(result,10,modulus)*mod_power(base,digit,modulus)%modulus
    return result

Run and test the lab

Copy the implementation and these assertions into a Python 3 file and run it. An assertion failure means the actual result differs from the expectation. Add the current topic's exercise and boundary tests to deepen your understanding.

assert gcd(48,18)==6 and gcd(-8,0)==8 and gcd(0,0)==0
assert mod_power(3,5,7)==pow(3,5,7)==5
assert mod_power(9,0,1)==0
a=[[1,2],[3,4]]; rotate(a); assert a==[[3,1],[4,2]]
assert all(mod_power(5,e,13)==pow(5,e,13) for e in range(30))
assert count_primes(10)==4 and count_primes(0)==0
assert count_primes(2)==0 and count_primes(100)==25
assert digit_power(2,[1,0],1337)==pow(2,10,1337)
assert digit_power(3,[2,0,1],1337)==pow(3,201,1337)