The Ugly Number Problem¶
This is a Conversation with Claude 4.0 about the solution to the Ugly Number Problem.
PROMPT¶
Can you add mathematical expressions for this explanation :
Solution for : https://leetcode.com/problems/ugly-number/
Problem Statement: An ugly number is a positive integer which does not have a prime factor other than 2, 3, and 5.Given an integer n, return true if n is an ugly number.
This is a great number theory problem, The fundamental theorem of arithmetic, also callled the prime factorization theorem states that any number greater than 1 is either a prime number or can be expressed as a product of prime numbers (ignoring the order).
12 = 2 * 2 * 3 = 2**2 * 3
210 = 2 * 3 * 5 * 7
By definition of ugnliness as stated in the problem, 12 is ugly but 210 is not because it has 7 as a factor. To test ugliness, all you have to do is remove all the factors of 2,3 and 5 and assert that only 1 remains.
But, what does it mean to remove, say ONE factor of 2 from some number x. It means, that the number x has is made up by adding some other number twice x = x/2 + x/2. We can test this by checking if 2 divides x evenly i.e. x%2 = 0 Removing one factor of two means removing one copy of x/2 from x. We keep doing this until x%2 != 0, to remove all factors of 2. Of the remaining number y, it can be said that there is no other number which if added to itself will yield y.
Similarly, for 3, after removing all factors of 3 from x, of the remaining number y, it can be said that there is no other number which if added to itself thrice will yield y.
RESPONSE¶
Here's the solution with mathematical expressions added to make the concepts clearer:
Solution for Ugly Number Problem¶
Problem Statement: An ugly number is a positive integer which does not have a prime factor other than 2, 3, and 5.
Mathematical Foundation¶
The Fundamental Theorem of Arithmetic states that every integer n > 1 can be uniquely expressed as:
n = p₁^a₁ × p₂^a₂ × p₃^a₃ × ... × pₖ^aₖ
where p₁, p₂, ..., pₖ are distinct prime numbers and a₁, a₂, ..., aₖ are positive integers.
Examples:
- 12 = 2² × 3¹ (ugly - only primes 2 and 3)
- 210 = 2¹ × 3¹ × 5¹ × 7¹ (not ugly - contains prime 7)
Definition of Ugliness¶
A number n is ugly if and only if:
n = 2^a × 3^b × 5^c
a, b, c ≥ 0 are non-negative integers.
Algorithm: Factor Removal¶
To test ugliness, we systematically remove all factors of 2, 3, and 5:
Step 1: Remove all factors of 2
While n ≡ 0 (mod 2):
n ← n/2
n ≢ 0 (mod 2), meaning no more factors of 2 remain.
Step 2: Remove all factors of 3
While n ≡ 0 (mod 3):
n ← n/3
Step 3: Remove all factors of 5
While n ≡ 0 (mod 5):
n ← n/5
Mathematical Proof of Correctness¶
After removing all factors of 2, 3, and 5, we have:
n_final = n_original / (2^a × 3^b × 5^c)
Case 1: If n_original was ugly, then n_original = 2^a × 3^b × 5^c, so:
n_final = (2^a × 3^b × 5^c) / (2^a × 3^b × 5^c) = 1
Case 2: If n_original was not ugly, then n_original = 2^a × 3^b × 5^c × p₁^d₁ × ... × pₖ^dₖ where pᵢ ∉ {2,3,5}, so:
n_final = (2^a × 3^b × 5^c × p₁^d₁ × ... × pₖ^dₖ) / (2^a × 3^b × 5^c) = p₁^d₁ × ... × pₖ^dₖ > 1
Modular Arithmetic Interpretation¶
When we check n % 2 == 0, we're testing the congruence:
n ≡ 0 (mod 2)
This is equivalent to asking: "Does there exist an integer k such that n = 2k?"
If yes, then 2 is a factor, and we can "remove" it by setting n ← k = n/2.
Implementation¶
def isUgly(n):
"""
Determines if n is an ugly number.
Mathematical approach: n is ugly iff n = 2^a × 3^b × 5^c for some a,b,c ≥ 0
Args:
n: integer to test
Returns:
bool: True if n is ugly, False otherwise
"""
# Base case: ugly numbers must be positive
if n <= 0:
return False
# Remove all factors of 2: n ← n/2 while n ≡ 0 (mod 2)
while n % 2 == 0:
n //= 2
# Remove all factors of 3: n ← n/3 while n ≡ 0 (mod 3)
while n % 3 == 0:
n //= 3
# Remove all factors of 5: n ← n/5 while n ≡ 0 (mod 5)
while n % 5 == 0:
n //= 5
# If n = 2^a × 3^b × 5^c originally, then n = 1 now
# If n had other prime factors, then n > 1 now
return n == 1
Implementation### Time and Space Complexity¶
Time Complexity: O(log n)
- In the worst case, we divide by 2 approximately
log₂(n)times - Similarly for factors of 3 and 5
- Total:
O(log₂(n) + log₃(n) + log₅(n)) = O(log n)
Space Complexity: O(1)
- Only using a constant amount of extra space
The mathematical foundation shows that this algorithm correctly identifies ugly numbers by leveraging the unique prime factorization and systematically removing the allowed prime factors through modular arithmetic operations.
PROMPT¶
Explain this notation :n ← n/5 while n ≡ 0 (mod 5)
RESPONSE¶
Excellent! Let me break down this chain of mathematical equivalences step by step:
The Triple Equivalence¶
n ≡ 0 (mod 5) ⟺ n % 5 == 0 ⟺ ∃k ∈ ℤ such that n = 5k
The symbol ⟺ means "if and only if" - these three statements are completely equivalent and mean exactly the same thing.
Statement 1: n ≡ 0 (mod 5)¶
Reading: "n is congruent to 0 modulo 5"
Mathematical Definition: Two integers are congruent modulo m if they have the same remainder when divided by m.
Since we're comparing n to 0:
- n ≡ 0 (mod 5) means n and 0 have the same remainder when divided by 5
- The remainder when 0 is divided by 5 is 0
- Therefore: n has remainder 0 when divided by 5
Statement 2: n % 5 == 0¶
Reading: "n modulo 5 equals 0" or "the remainder when n is divided by 5 is 0"
Programming Context: This is how we write modular arithmetic in most programming languages.
Examples:
- 10 % 5 = 0 ✓
- 15 % 5 = 0 ✓
- 17 % 5 = 2 ✗
- 23 % 5 = 3 ✗
Statement 3: ∃k ∈ ℤ such that n = 5k¶
Reading: "There exists an integer k such that n equals 5 times k"
Symbol Breakdown:
∃= "there exists"k ∈ ℤ= "k is an element of the integers" (k is a whole number: ..., -2, -1, 0, 1, 2, ...)n = 5k= "n equals 5 times k"
Meaning: n can be written as some integer multiplied by 5 (i.e., n is a multiple of 5)
Why These Are All Equivalent¶
Let me prove these equivalences with examples:
Example 1: n = 20¶
Statement 1: 20 ≡ 0 (mod 5)
- 20 ÷ 5 = 4 remainder 0 ✓
Statement 2: 20 % 5 == 0
- 20 % 5 = 0 ✓
Statement 3: ∃k ∈ ℤ such that 20 = 5k
- k = 4 works: 20 = 5 × 4 ✓
Example 2: n = 17¶
Statement 1: 17 ≡ 0 (mod 5)
- 17 ÷ 5 = 3 remainder 2 ≠ 0 ✗
Statement 2: 17 % 5 == 0
- 17 % 5 = 2 ≠ 0 ✗
Statement 3: ∃k ∈ ℤ such that 17 = 5k
- No integer k works: 17/5 = 3.4 (not an integer) ✗
The Mathematical Connection¶
Division Algorithm¶
For any integers n and 5, we can write:
n = 5q + r where 0 ≤ r < 5
The Three Statements Are Saying:¶
- Modular: The remainder r equals 0
- Programming: The remainder operation gives 0
- Algebraic: We can write n as 5 times some integer (when r = 0, then n = 5q)
Practical Examples¶
| n | n ÷ 5 | Quotient (q) | Remainder (r) | n ≡ 0 (mod 5)? | n % 5 | k exists? | n = 5k |
|---|---|---|---|---|---|---|---|
| 0 | 0 ÷ 5 | 0 | 0 | ✓ | 0 | ✓ (k=0) | 0 = 5×0 |
| 5 | 5 ÷ 5 | 1 | 0 | ✓ | 0 | ✓ (k=1) | 5 = 5×1 |
| 10 | 10 ÷ 5 | 2 | 0 | ✓ | 0 | ✓ (k=2) | 10 = 5×2 |
| 12 | 12 ÷ 5 | 2 | 2 | ✗ | 2 | ✗ | 12 ≠ 5k |
| -15 | -15 ÷ 5 | -3 | 0 | ✓ | 0 | ✓ (k=-3) | -15 = 5×(-3) |
Key Insight¶
All three expressions are different ways of asking the same fundamental question:
"Is n a multiple of 5?"
- Modular arithmetic asks: "Does n have remainder 0 when divided by 5?"
- Programming asks: "Does the modulo operation return 0?"
- Algebraic asks: "Can we write n as 5 times some integer?"
They're all equivalent ways of expressing divisibility!