Divide Integers Without Division

MediumPython

Problem

The Fannie Mae Desktop Underwriter (DU) calculation layer needs integer division for bounded values, but the arithmetic primitive cannot use division, modulo, or multiplication operators. Implement a function that returns the quotient and remainder of two integers.

The quotient must truncate toward zero. The remainder must have the same sign as the dividend, satisfying dividend = divisor * quotient + remainder mathematically. You may use addition, subtraction, comparisons, bit shifts, bitwise operations, and unary negation. The divisor is guaranteed to be nonzero.

Formal Specification

Implement divide_integers(dividend, divisor).

Input: Two integers, dividend and divisor.
Output: A two-element list [quotient, remainder].
Do not use /, //, %, or multiplication in the implementation.
The result should support signed 32-bit input values and mathematical integer output.

Constraints

-2^31 <= dividend <= 2^31 - 1
-2^31 <= divisor <= 2^31 - 1
divisor != 0
Do not use /, //, %, or multiplication
Return mathematical integer results without fixed-width overflow

Function Signature

def divide_integers(dividend, divisor):

Formal Specification

Implement divide_integers(dividend, divisor).

Input: Two integers, dividend and divisor.
Output: A two-element list [quotient, remainder].
Do not use /, //, %, or multiplication in the implementation.
The result should support signed 32-bit input values and mathematical integer output.