Division
Division Algorithm
The Division Algorithm is a fundamental theorem in mathematics that states: for any two integers a (dividend) and n (divisor, where n > 0), there exist unique integers q (quotient) and r (remainder) such that a = nq + r, where 0 ≤ r < n. This algorithm guarantees that division always results in a quotient and remainder that satisfy these conditions.
Key Components:
( a = nq + r )
1. Dividend (a): The number being divided
2. Divisor (n): The number we divide by (must be positive)
3. Quotient (q): How many times n goes into a
4. Remainder (r): What's left over (must be less than n)
Key Rules:
• Remainder must be non-negative (r ≥ 0)
• Remainder must be less than divisor (r < n)
• Result must equal dividend (a = nq + r)
• Solution must be unique
Example 1:
Let a = 17, n = 5
Process:
• Find largest q where 5q ≤ 17 = 3
• q = 3 (as 5 × 3 = 15)
• r = 17 - 15 = 2 Therefore: 17 = 5(3) + 2
Example 2:
Let a = -13, n = 4
Process:
• Find q where remainder is positive
• q = -4 (as -13 = 4(-4) + 3)
• r = 3 Therefore: -13 = 4(-4) + 3