Ugly Number

An ugly number is a positive integer whose only prime factors are 2, 3, or 5. To determine whether a given integer ( n ) is ugly, we can repeatedly divide it by these prime factors as long as it is divisible by them. If we are left with 1, then the number is ugly.

Here’s the code:

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class Solution:
    def isUgly(self, num: int) -> bool:
        if num <= 0:
            return False

        # Divide num by 2, 3, and 5 as long as it is divisible by them
        for prime_factor in [2, 3, 5]:
            while num % prime_factor == 0:
                num = num // prime_factor

        # If num has become 1, then it is an ugly number
        return num == 1

Key Insights

  • We continuously divide the number by 2, 3, and 5, as long as it is divisible by them.
  • If we end up with 1, then the original number was composed only of these prime factors, and it is an ugly number.
  • If the number is less than or equal to 0, it’s not considered ugly, so we return False immediately.

Identifying Problem Isomorphism

“Ugly Number”: “Happy Number”

In “Ugly Number”, you are given an integer n and you are supposed to determine whether n is an “ugly” number. An “ugly” number is defined as a positive number whose prime factors only include 2, 3, and 5.

Similarly, in “Happy Number”, you are given an integer n, and you need to determine whether n is a “happy” number. A “happy” number is defined as a number which eventually reaches 1 when replaced by the sum of the square of each digit.

In both problems, you are examining the properties of a given integer and determining whether it fits into a specific category (“ugly” or “happy”). You can solve both problems by repeatedly transforming the number according to some rule and checking if the result fits the condition.

However, these problems handle very different transformations and conditions. While “Ugly Number” involves prime factorization, “Happy Number” involves digit manipulation. This makes the problems different in nature, but they still share the fundamental idea of evaluating whether a number fits into a certain class based on some transformations.

“Ugly Number” is simpler because prime factorization and checking if a number is divisible by 2, 3 or 5 is a straightforward task. In contrast, the process to check if a number is “happy” involves more complex steps like splitting the number into digits, squaring each digit, and summing up the squares, which makes it more complex.

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class Solution:
    def isUgly(self, num: int) -> bool:
        if num == 0: return False
        while num % 5 == 0: num /= 5
        while num % 3 == 0: num /= 3
        while num % 2 == 0: num /= 2
        return num == 1

Problem Classification

This problem can be classified as:

  1. Number Theory: This is the most prominent domain of this problem. The problem is primarily about the factors of a number, particularly its prime factors. Understanding the fundamentals of number theory, including the concept of prime factors, is crucial to solving this problem.

  2. Decision Problems: This problem is a decision problem where the solution is either true or false based on whether the given number satisfies a certain condition.

  3. Arithmetic: Basic arithmetic operations are involved in the process of finding the prime factors.

  4. Iterative/Recursive Problem: To solve this problem, we have to repeatedly divide the given number by its prime factors until we cannot divide it anymore. This involves a concept of loop or recursion.

To solve this problem, the key concept is to repeatedly divide the given number by its prime factors (2, 3, 5) if it is divisible. If the remaining number after all such divisions is 1, then the number is an ugly number. Otherwise, it is not an ugly number.

The learning units can be identified as follows:

  1. Understanding prime numbers and prime factors
  2. Understanding and implementing decision problems in code
  3. Basic arithmetic operations
  4. Understanding and implementing loops or recursive functions
  5. Implementing the process of dividing a number by its factors
  6. Integrating the above concepts to solve the problem.

Language Agnostic Coding Drills

Breaking down the problem-solving process for this problem, we get the following steps:

  1. Problem Analysis: Understand the problem statement and requirements. An ugly number is one whose prime factors are only 2, 3, and 5. We need to return a boolean value, true if the number is ugly and false otherwise.

  2. Edge Case Handling: For the number to be ugly, it needs to be a positive integer. Therefore, if the given number is 0, we should return false as it does not meet the definition of an ugly number.

  3. Division by Prime Factors: The main concept in the problem is to keep dividing the given number by its prime factors until we cannot divide it anymore. This is done in three steps:

    • While the number is divisible by 5, keep dividing it by 5.
    • Once it’s no longer divisible by 5, while it’s divisible by 3, keep dividing it by 3.
    • Finally, while the number is divisible by 2, keep dividing it by 2.

  4. Result Evaluation: At the end of the divisions, if the resulting number is 1, then the original number is an ugly number, so we return true. Otherwise, it’s not an ugly number and we return false.

The learning units for this problem can be listed as follows:

  1. Understanding and identifying prime factors: Recognize that 2, 3, and 5 are the only prime factors for an ugly number.

  2. Conditional statements and returning boolean values: Learn to return boolean values based on a condition (In this case, whether or not the number is an ugly number).

  3. Loop control structures and division operations: Implement loops that continue as long as the number is divisible by a certain factor, and perform the division operation within the loop.

  4. Edge case handling: Handle the case where the input number is zero.

  5. Integration of the above concepts: Combine all the above concepts to solve the problem and identify an ugly number.

Targeted Drills in Python

Drill 1: Understanding and identifying prime factors Create a function to identify whether a number is a prime number.

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def is_prime(n):
    if n < 2:
        return False
    for i in range(2, int(n**0.5) + 1):
        if n % i == 0:
            return False
    return True

Drill 2: Conditional statements and returning boolean values Learn to return boolean values based on a condition. Here, we will create a function that returns True if the number is even and False otherwise.

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def is_even(n):
    return n % 2 == 0

Drill 3: Loop control structures and division operations Implement loops that continue as long as the number is divisible by a certain factor, and perform the division operation within the loop.

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def divide_by_factor(n, factor):
    while n % factor == 0:
        n /= factor
    return n

Drill 4: Edge case handling Handle the case where the input number is zero.

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def check_zero(n):
    return n != 0

Drill 5: Integration of the above concepts Combine all the above concepts to solve the problem and identify an ugly number.

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def is_ugly(num):
    if not check_zero(num):
        return False
    for i in [2, 3, 5]:
        num = divide_by_factor(num, i)
    return num == 1

Each of the above drills can be practiced separately and then combined to create the final solution. The drills start with simple concepts and build up to the final solution, giving you a step-by-step approach to the problem.