The remainder in division must be less than the divisor because that ensures a unique and proper quotient. If the remainder were as large as or larger than the divisor, more division could be performed, yielding a different, smaller remainder and a larger quotient. This guarantees a single, well-defined result for each division problem. Key points:

• In integer division, you seek numbers q and r such that: dividend = divisor × q + r, with 0 ≤ r < |divisor|.
• If r ≥ |divisor|, you can increase q by 1 and subtract another divisor, producing a smaller r and a larger q. This shows the current r wasn’t the true remainder.
• For negative divisors, the standard convention still aims for a remainder whose absolute value is less than the absolute value of the divisor, and r's sign typically matches the dividend in many definitions.

Illustrative intuition:

• Example with positive divisor: 83 divided by 10 yields q = 8 and r = 3, since 83 = 10×8 + 3 and 0 ≤ 3 < 10. If r were 13, you could have increased q to 9 and subtracted another 10, which would contradict the chosen remainder.
• Example with negative divisor: 83 divided by -10 would yield q = -8 and r = 3 in the common convention, still keeping 0 ≤ r < |divisor| = 10.

Why it matters:

• Uniqueness: The condition 0 ≤ r < |divisor| ensures every dividend has exactly one pair (q, r) that satisfies the equation.
• Consistency: This standard underpins arithmetic of integers, modular arithmetic, and algorithms that rely on clear quotient and remainder.

If you’d like, provide a specific division problem, and it can be worked through step-by-step to illustrate how the remainder remains strictly less than the divisor.