Divide Two Integers
To divide two integers without using multiplication, division, or the mod operator, you can use the technique of repeated subtraction and doubling. Here’s how you can do it:
 Determine the Sign of the Result: First, determine the sign of the result by checking the signs of both the dividend and divisor.
 Work with Positive Values: Convert the dividend and divisor to their absolute values, as it’s easier to work with positive numbers.
 Repeated Subtraction and Doubling: Start by subtracting the divisor from the dividend. Double the divisor and the corresponding quotient as long as the doubled divisor is less than or equal to the dividend.
 Repeat the Process: Repeat the above step until the dividend is less than the original divisor.
 Apply the Sign: Apply the previously determined sign to the result.
 Handle Overflow: Check if the result is within the 32bit signed integer range. If not, return the corresponding boundary value.
Here’s a Python function that implements the above steps:


This function will return the quotient of the division, truncated towards zero. The time complexity is (O(\log(\text{{dividend}}/\text{{divisor}}))), and the space complexity is (O(1)).
Can you help me with finding the isomorphism for the LeetCode ? Please provide a concise response and if an exact mapping is not possible, specify it as an approximate mapping. If the titles of the problems are the same, select a different problem. Also, explain the reasoning behind your selection. Don’t mention anything about your training cut off date. Provide the best possible answer. Do not repeat the same point. Mention which problem is simpler. If there are multiple isomorphic problem. Arrange them from simple to more complex to your best possible knowledge. Do not add Leetcode to the problem name. The response format is:
The problem can be mapped to


Problem Classification
Given two integers dividend and divisor, divide two integers without using multiplication, division, and mod operator.
The integer division should truncate toward zero, which means losing its fractional part. For example, 8.345 would be truncated to 8, and 2.7335 would be truncated to 2.
Return the quotient after dividing dividend by divisor.
Note: Assume we are dealing with an environment that could only store integers within the 32bit signed integer range: [−231, 231 − 1]. For this problem, if the quotient is strictly greater than 231  1, then return 231  1, and if the quotient is strictly less than 231, then return 231.
Example 1:
Input: dividend = 10, divisor = 3 Output: 3 Explanation: 10/3 = 3.33333.. which is truncated to 3.
Example 2:
Input: dividend = 7, divisor = 3 Output: 2 Explanation: 7/3 = 2.33333.. which is truncated to 2.
Constraints:
231 <= dividend, divisor <= 231  1 divisor != 0
Identifying Problem Isomorphism
Can you help me with finding the isomorphism for this problem?
Which problem does this problem map to the corresponding isomorphic problem on Leetcode ?
How did you identify that this problem is a variation of problem?
Language Agnostic Coding Drills
Boolean and comparison operations: Understanding how boolean and comparison operations work is fundamental in every programming language. Here, in line 1, we see a check for whether the sign of
dividend
is the same asdivisor
. If they’re the same sign,positive
isTrue
. If they’re different signs,positive
isFalse
.Builtin functions, including absolute value (
abs
): Builtin functions are tools provided by the language to help with common tasks.abs
is one such function, used here to makedividend
anddivisor
both nonnegative.Variable assignment: Basic concept that covers how to assign a value to a variable. Here,
res
is set to 0 initially, and will later hold the result of the division operation.Looping constructs: Understanding how to use
while
loops is critical. The condition for the outerwhile
loop isdividend >= divisor
.Simultaneous multiple assignment: This involves assigning values to multiple variables in a single line. In this line,
curr_divisor
is set todivisor
andnum_divisors
is set to1
.Nested looping constructs: The concept of having loops inside other loops (nested loops) is used in this code.
Basic arithmetic operations and variable reassignment: Here,
dividend
is reduced bycurr_divisor
and the result is reassigned todividend
.Variable incrementation:
res
is increased bynum_divisors
.Bitwise operations: Here the
<<
operator is used, which shifts the bits ofcurr_divisor
one bit to the left, effectively doubling the value ofcurr_divisor
.Bitwise operations: Similar to step 9,
num_divisors
is also doubled here.Conditional Statements: Understanding how to use
if
statements to run code based on whether a condition isTrue
.Unary operations:
res
is negated ifpositive
isFalse
.Builtin function  min and max:
min
andmax
are used here to ensure the result is within the range of a 32bit signed integer.
In terms of a problemsolving approach, the code is effectively implementing a form of division using bit manipulation to increase the efficiency. The bitwise left shift operation doubles the value of curr_divisor
and num_divisors
each time through the inner loop, making it a more efficient solution for large dividends. The result is kept within the bounds of a 32bit signed integer by using the min
and max
functions.
Targeted Drills in Python
Boolean and comparison operations:
Problem: Given two integers,
a
andb
, returnTrue
if they have the same sign andFalse
otherwise.1 2
def same_sign(a, b): return (a < 0) is (b < 0)
Builtin functions, including absolute value (
abs
):Problem: Given an integer list, create a new list with absolute values of all elements from the original list.
1 2
def absolute_list(nums): return [abs(num) for num in nums]
Variable assignment:
Problem: Create a function that returns the sum of two numbers.
1 2 3
def add(a, b): sum = a + b return sum
Looping constructs:
Problem: Create a function that returns the sum of all elements in a list.
1 2 3 4 5
def sum_list(nums): total = 0 for num in nums: total += num return total
Simultaneous multiple assignment:
Problem: Create a function that returns two numbers swapped.
1 2 3
def swap(a, b): a, b = b, a return a, b
Nested looping constructs:
Problem: Given a 2D list, return the sum of all elements.
1 2 3 4 5 6
def sum_matrix(matrix): total = 0 for row in matrix: for element in row: total += element return total
Basic arithmetic operations and variable reassignment:
Problem: Subtract a value from a given number and return the result.
1 2 3
def subtract(n, val): n = val return n
Variable incrementation:
Problem: Create a function that adds one to a given number.
1 2 3
def increment(n): n += 1 return n
Bitwise operations:
Problem: Given a number, return the number after left shifting its bits by one place.
1 2
def left_shift(n): return n << 1
Conditional Statements:
Problem: Given two numbers, return the greater one.
1 2 3 4 5
def max_num(a, b): if a > b: return a else: return b
Unary operations:
Problem: Given a number, return its negation.
1 2
def negate(n): return n
Builtin function  min and max:
Problem: Given three numbers, return the middle one.
1 2
def middle_num(a, b, c): return max(min(a, b), min(max(a, b), c))
Each of these drills focuses on a specific concept and can be used to help students understand how each piece of the overall problem is solved. When combined, these concepts form the basis of the solution to the overall problem.
10 Prerequisite LeetCode Problems
The “29. Divide Two Integers” problem asks you to divide two integers without using multiplication, division, and mod operators. Here are some simpler problems that help build the understanding and skills required to solve this problem:
7. Reverse Integer: This problem is an excellent introduction to integer manipulation in programming languages.
258. Add Digits: This problem helps with understanding how to operate with digits of an integer.
9. Palindrome Number: Like Reverse Integer, this problem continues to explore integer manipulation and understanding the properties of numbers.
231. Power of Two: Helps understand concepts around exponents, which will be useful in the “Divide Two Integers” problem.
1342. Number of Steps to Reduce a Number to Zero: This problem helps with understanding stepwise operations on an integer.
191. Number of 1 Bits: This problem introduces bit manipulation, a powerful tool for this problem.
371. Sum of Two Integers: Further builds upon the understanding of bit manipulation.
136. Single Number: Another problem that introduces XOR operation and its properties.
190. Reverse Bits: This problem explores more indepth bit manipulation, which will help understand the bitlevel operations.
338. Counting Bits: Another good problem for understanding and working with bits.
These problems will help you understand integer and bit manipulation in a programming language and how to use these techniques to solve problems without using traditional mathematical operators.
Problem Analysis and Key Insights
What are the key insights from analyzing the problem statement?
Problem Boundary
What is the scope of this problem?
How to establish the boundary of this problem?
Problem Classification
Problem Statement:Given two integers dividend and divisor, divide two integers without using multiplication, division, and mod operator.
The integer division should truncate toward zero, which means losing its fractional part. For example, 8.345 would be truncated to 8, and 2.7335 would be truncated to 2.
Return the quotient after dividing dividend by divisor.
Note: Assume we are dealing with an environment that could only store integers within the 32bit signed integer range: [−231, 231 − 1]. For this problem, if the quotient is strictly greater than 231  1, then return 231  1, and if the quotient is strictly less than 231, then return 231.
Example 1:
Input: dividend = 10, divisor = 3 Output: 3 Explanation: 10/3 = 3.33333.. which is truncated to 3.
Example 2:
Input: dividend = 7, divisor = 3 Output: 2 Explanation: 7/3 = 2.33333.. which is truncated to 2.
Constraints:
231 <= dividend, divisor <= 231  1 divisor != 0
Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.
Distilling the Problem to Its Core Elements
Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?
Visual Model of the Problem
How to visualize the problem statement for this problem?
Problem Restatement
Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.
Abstract Representation of the Problem
Could you help me formulate an abstract representation of this problem?
Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific realworld details?
Terminology
Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?
Problem Simplification and Explanation
Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?
Constraints
Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.
What are the key insights from analyzing the constraints?
Case Analysis
Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.
Provide names by categorizing these cases
What are the edge cases?
What are the key insights from analyzing the different cases?
Identification of Applicable Theoretical Concepts
Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?
Simple Explanation
Can you explain this problem in simple terms or like you would explain to a nontechnical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?
Problem Breakdown and Solution Methodology
Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.
Inference of ProblemSolving Approach from the Problem Statement
Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method.
How did you infer from the problem statement that this problem can be solved using ?
Simple Explanation of the Proof
I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?
Stepwise Refinement
Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the highlevel solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?
Solution Approach and Analysis
Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.
Identify Invariant
What is the invariant in this problem?
Identify Loop Invariant
What is the loop invariant in this problem?
Thought Process
Can you explain the basic thought process and steps involved in solving this type of problem?
Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.
Establishing Preconditions and Postconditions
Parameters:
 What are the inputs to the method?
 What types are these parameters?
 What do these parameters represent in the context of the problem?
Preconditions:
 Before this method is called, what must be true about the state of the program or the values of the parameters?
 Are there any constraints on the input parameters?
 Is there a specific state that the program or some part of it must be in?
Method Functionality:
 What is this method expected to do?
 How does it interact with the inputs and the current state of the program?
Postconditions:
 After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
 What does the return value represent or indicate?
 What side effects, if any, does the method have?
Error Handling:
 How does the method respond if the preconditions are not met?
 Does it throw an exception, return a special value, or do something else?
Problem Decomposition
Problem Understanding:
 Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
 Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
 For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
 Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
 For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
 Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
 How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?
From Brute Force to Optimal Solution
Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?
Code Explanation and Design Decisions
Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?
Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.
Coding Constructs
Consider the following piece of complex software code.
What are the highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?
Language Agnostic Coding Drills
Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.
Dissect the code and identify each distinct concept it contains. Remember, this process should be languageagnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problemsolving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the stepbystep process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.
Targeted Drills in Python
Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Pythonbased coding drills for each of those concepts.
Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problemspecific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.
Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.
Q&A
Similar Problems
Can you suggest 10 problems from LeetCode that require similar problemsolving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem.