Problem 54 - Fundamentals of Mathematics

Question

What number is the smallest prime number?

Step-by-Step Solution

Answer

The smallest prime number is 2.

Step 1: Understanding Prime Numbers

A prime number is a natural number greater than 1 that has no divisors other than 1 and itself. To be classified as prime, a number must not be divisible evenly by any other numbers.

Step 2: Identifying the Smallest Prime

We begin examining numbers from 2, as 1 is not considered a prime number because a prime number must have exactly two distinct divisors. Number 2 itself has divisors of 1 and 2 only, fulfilling the criteria of being a prime number.

Key Concepts

Prime numbers

Prime numbers are incredibly significant in mathematics due to their unique properties. They are the building blocks of all natural numbers, much like atoms are in chemistry. A prime number is any natural number greater than 1 that is divisible only by 1 and itself. For example, 2, 3, 5, 7, and 11 are all prime numbers.
When identifying prime numbers:

• The number should only have two divisors.
• It cannot be evenly divided by any other numbers.

One important aspect to note is that the number 2 is the only even prime number. Every other even number can be divided by 2, making them non-prime. This highlights just how special the number 2 is within the set of natural numbers. Every other prime number will be odd.

Natural numbers

Natural numbers are the simplest numbers we use in our daily lives and fundamental to mathematics. They are numbers that start from 1 and continue indefinitely, such as 1, 2, 3, 4, and so on. In mathematical terms,

• Natural numbers are positive and whole numbers.
• They make up the set often denoted by the symbol \( \mathbb{N} \).

The concept of natural numbers is pivotal when discussing prime numbers because all prime numbers are also natural numbers. However, it's important to note that not all natural numbers are prime. A number needs to meet the specific criteria of having only two distinct divisors to be classified as a prime number.

Divisors

Divisors play a key role in understanding numbers, especially when it comes to primes. A divisor of a number is an integer that can divide the number without leaving a remainder. For instance, for the number 6, its divisors are 1, 2, 3, and 6, because each can evenly divide 6.

• A prime number will have exactly two divisors.
• These are always 1 and the number itself.

When determining if a number is prime, we check it for divisors beyond 1 and itself. If any extra divisors are found, the number is not prime. Understanding divisors is crucial for identifying and working with prime numbers, as they define the very criteria of what makes a number prime.