Problem 45 - Fundamentals of Mathematics

Question

Determine which of the whole numbers are prime and which are composite. 25

Step-by-Step Solution

Answer

25 is a composite number.

Step 1: Define Prime and Composite Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a natural number greater than 1 that has more than two positive divisors.

Step 2: Identify Divisors of 25

Check the divisors of 25. A divisor of a number divides that number without leaving a remainder.

Step 3: List Divisors of 25

The divisors of 25 are 1, 5, and 25. This shows that 25 can be divided by 1, itself, and another number, namely 5.

Step 4: Determine Primality

Since 25 has more than two divisors, it is not a prime number. It fits the definition of a composite number.

Key Concepts

Composite Numbers

Composite numbers are an essential concept in mathematics. They are natural numbers that have more than two distinct positive divisors. This means, unlike prime numbers, composite numbers can be divided exactly by numbers other than just 1 and themselves. For example, the number 25 is a composite number because it has three divisors: 1, 5, and 25.
Understanding composite numbers is critical because identifying them helps us in numerous mathematical operations including factorization.
Composite numbers can be expressed as the product of smaller natural numbers, which we often refer to as its prime factors. Take 25 again; it can be expressed as 5 multiplied by 5, or simply, 5 squared. This feature makes composite numbers particularly useful in breaking down complex problems into simpler parts.

Natural Numbers

Natural numbers are the basic building blocks of mathematics. These are the numbers you naturally start counting with: 1, 2, 3, and so on. They do not include zero, negative numbers, or fractions. Natural numbers are instrumental in a wide range of mathematical concepts, including the classification of numbers into prime and composite.
One significant character of natural numbers is that they are the simplest numbers used in everyday counting and ordering. Additionally, they are used to define divisors.
Being familiar with natural numbers is foundational when studying more complex mathematical concepts, as almost all number theories begin with understanding the properties of natural numbers.

Divisors

Divisors, sometimes referred to as factors, are numbers that divide another number completely, without leaving a remainder. For instance, if you can divide a number "A" by another number "B" and the result is a whole number, then "B" is a divisor of "A".
Consider the number 25 again. If we look at its divisors, we find that 1, 5, and 25 itself are divisors of 25 because dividing 25 by these numbers leaves no remainder:

• 25 ÷ 1 = 25
• 25 ÷ 5 = 5
• 25 ÷ 25 = 1

Divisors play an important role in determining whether a number is prime or composite. If a number has exactly two divisors, which are 1 and itself, it is a prime number. If it has more than two divisors, it is a composite number.