Problem 12 - Fundamentals of Mathematics

Question

Determine which of the following whole numbers are prime and which are composite. 47

Step-by-Step Solution

Answer

47 is a prime number.

Step 1: Introduction

To determine if a number is prime or composite, check if it has any divisors other than 1 and itself. If it does, it's composite; if it does not, then it is prime.

Step 1: Check Divisibility by 2

Number 47 is not divisible by 2, as it is an odd number. Therefore, it cannot have 2 as a factor.

Step 2: Check Divisibility by 3

Add the digits of the number 47: 4 + 7 = 11. Since 11 is not divisible by 3, 47 also cannot be divided by 3 without a remainder.

Step 3: Check Divisibility by 5

Since 47 does not end in 0 or 5, it is not divisible by 5.

Step 4: Check Divisibility by 7

Dividing 47 by 7, we get approximately 6.71, which is not a whole number. Therefore, 47 is not divisible by 7.

Step 6: Conclusion

No whole numbers except 1 and 47 can divide 47 without leaving a remainder, indicating that 47 is a prime number.

Key Concepts

Composite Numbers

Composite numbers are an essential part of number theory. They are natural numbers that have more than two distinct positive divisors. In simpler terms, a composite number can be divided evenly by numbers other than 1 and itself. For instance, the number 6 is composite because it is divisible by 1, 2, 3, and 6.

Understanding composite numbers helps in identifying prime numbers. They help form the basis of factorization, where numbers are broken down into products of lower numbers. If a number isn't prime, it is usually composite. But simply knowing a number is composite isn't enough; knowing its factors can be very helpful.

To find out if a number is composite, try dividing it by smaller prime numbers like 2, 3, 5, or 7, depending on the size of the number. If any of these can divide the number evenly, without a remainder, then that number is composite.

Divisibility Rules

Divisibility rules are simple shortcuts to determine whether a number can be divided by another number without actually carrying out the division. These rules provide a quick way to ascertain a number's divisibility, making it easier to determine if it's prime or composite.

• Divisibility by 2: A number is divisible by 2 if it is even, meaning its last digit is 0, 2, 4, 6, or 8.
• Divisibility by 3: If the sum of a number’s digits is divisible by 3, then the number itself is divisible by 3.
• Divisibility by 5: A number is divisible by 5 if it ends in 0 or 5.
• Divisibility by 7: For 7, doubling the last digit and subtracting it from the rest of the number can tell you if it divides evenly by 7.

These rules are incredibly useful when trying to quickly determine whether a number is a candidate for being prime.

Number Theory

Number theory is a branch of mathematics that delves deep into the properties and relationships between numbers, especially integers. This field explores various concepts such as divisibility, prime numbers, and equations involving integers.

At the heart of number theory are prime numbers and their unique qualities. Since every positive integer greater than 1 is either a prime or can be factored into prime numbers, understanding them is crucial for anyone studying mathematics.

Number theory involves learning about:

• The distribution of prime numbers
• Solutions to equations in integers
• Properties of whole numbers and their generalizations

The knowledge gained in number theory is not only fascinating but also essential in modern applications like cryptography.

Whole Numbers

Whole numbers form the basis of arithmetic and number theory. They include all non-negative integers, from 0 upwards, like 0, 1, 2, 3, and so on. Whole numbers are often utilized in counting and ordering.

These numbers are fundamental because they serve as the stepping stones to more complex number sets, like integers, which include negative numbers, and rational numbers, which include fractions. Whole numbers set the stage for understanding various mathematical operations, such as addition, subtraction, multiplication, and division.

The concept of whole numbers is also crucial for grasping the notion of primes and composites. In number theory, we often ask: Is a given whole number prime or composite? This distinction is the backbone of many mathematical explorations, including factorization and divisibility.