The concept of an “ugly number” may seem a bit unconventional, as we typically do not associate beauty or ugliness with numbers. However, in the context of mathematics, an ugly number refers to a specific category of numbers based on their prime factors. To understand what makes a number ugly, we need to delve into the world of prime factorization.
Prime factors are the prime numbers that can be multiplied together to obtain a given number. For example, the prime factors of 12 are 2 and 3, as 2 * 2 * 3 = 12. Similarly, the prime factors of 15 are 3 and 5, as 3 * 5 = 15.
Now, an ugly number is defined as a number whose prime factors are limited to 2, 3, and 5. In other words, if we can express a number as 2^a * 3^b * 5^c, where a, b, and c are non-negative integers, then it is considered an ugly number.
Let’s analyze the numbers from 1 to 15 to identify the ugly numbers within this range:
1: It is the smallest positive integer and has no prime factors at all. Since it does not have any prime factors, it is not considered an ugly number.
2: It is a prime number and has only one prime factor, which is 2. Since the prime factor is limited to 2, it qualifies as an ugly number.
3: It is also a prime number and has only one prime factor, which is 3. As the prime factor is limited to 3, it is an ugly number.
4: It can be expressed as 2^2, meaning it has only 2 as its prime factor. Thus, it is an ugly number.
5: Being a prime number, it has only one prime factor, which is 5. As it adheres to the limitation of prime factors, it is an ugly number.
6: It can be factored into 2 * 3, where both 2 and 3 are prime factors. Therefore, it falls under the category of ugly numbers.
7: It is a prime number and has no prime factors other than itself. Thus, it does not meet the criteria for being an ugly number.
8: It can be expressed as 2^3, meaning it has only 2 as its prime factor. Hence, it is an ugly number.
9: It can be factored into 3^2, having 3 as the only prime factor. Therefore, it qualifies as an ugly number.
10: It can be expressed as 2 * 5, where both 2 and 5 are prime factors. Thus, it is an ugly number.
11: It is a prime number and does not possess any prime factors other than itself. Hence, it is not an ugly number.
12: It can be factored into 2^2 * 3, having both 2 and 3 as prime factors. Thus, it is an ugly number.
13: It is a prime number and is only divisible by 1 and itself. Therefore, it does not meet the criteria for an ugly number.
14: It can be expressed as 2 * 7, where 7 is a prime factor. As it contains a prime factor other than 2, 3, and 5, it is not an ugly number.
15: It can be factored into 3 * 5, where both 3 and 5 are prime factors. Hence, it is an ugly number.
After analyzing the numbers from 1 to 15, we find that there are 11 ugly numbers within this range: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, and 15. The remaining numbers, namely 7, 11, 13, and 14, are not considered ugly numbers due to their prime factors falling outside the limited set of 2, 3, and 5.
The concept of an ugly number is based on the prime factors of a number, specifically limited to 2, 3, and 5. While the term “ugly” may seem subjective when applied to numbers, it serves as a useful classification in mathematics to categorize numbers with specific prime factorization patterns.