The Fundamental Theorem of Arithmetic

Introduction

In earlier classes, you’ve learned that any natural number can be represented as a product of its prime factors. For instance:

$\boldsymbol{2 = 2}$
$\boldsymbol{4 = 2 \times 2}$
$\boldsymbol{253 = 11 \times 23}$

This process of breaking down a number into prime factors is called prime factorization. The Fundamental Theorem of Arithmetic takes this idea further, stating that every composite number can be uniquely expressed as a product of prime factors, apart from the order of these factors.

Let’s explore this concept in more depth and examine how every natural number, whether small or large, can be represented using prime factors.

Prime Numbers and Their Role

Prime numbers, like 2, 3, 5, 7, and 11, are the building blocks of all natural numbers. If we take any set of prime numbers and multiply them in different ways, we can produce an infinite collection of natural numbers.

Examples of Multiplying Prime Numbers:

$\boldsymbol{7 \times 11 \times 23 = 1771}$
$\boldsymbol{3 \times 7 \times 11 \times 23 = 5313}$
$\boldsymbol{2 \times 3 \times 7 \times 11 \times 23 = 10626}$
$\boldsymbol{2^3 \times 3 \times 7^3 = 8232}$

These examples illustrate that by multiplying prime numbers in different combinations and allowing them to repeat as needed, we can generate an infinite collection of composite numbers. The Fundamental Theorem of Arithmetic confirms that every composite number can be created in this manner, ensuring that all possible products of primes result in composite numbers.

Statement of the Fundamental Theorem of Arithmetic

The theorem is formally stated as follows:

“Every composite number can be expressed (or factorized) as a product of primes, and this factorization is unique, apart from the order of factors.”

This means that for any composite number, there is only one way to write it as a product of primes if we disregard the order of the factors. For example, $\boldsymbol{2 \times 3 \times 5 \times 7}$ is the same as $\boldsymbol{3 \times 5 \times 7 \times 2}$ , as they represent the same prime factorization.

Unique Prime Factorization

The Fundamental Theorem of Arithmetic not only states that every composite number can be factorized into primes but also emphasizes uniqueness. This unique factorization means that no matter how we factorize a number, we will always end up with the same prime factors raised to the same powers.

Example of Unique Prime Factorization:

Factorize $\boldsymbol{32760}$ .
- Using a factor tree, we get: $\boldsymbol{32760 = 2^3 \times 3^2 \times 5 \times 7 \times 13}$ .

Similarly, factorizing $\boldsymbol{123456789}$ results in:

$\boldsymbol{123456789 = 3^2 \times 3803 \times 3607}$

where $\boldsymbol{3803}$ and $\boldsymbol{3607}$ are prime numbers.

This uniqueness is why prime factorization is so important: every composite number has a single “identity” when expressed as a product of primes.

Steps to Perform Prime Factorization

To find the prime factorization of a composite number:

Divide by the Smallest Prime: Start by dividing the number by the smallest prime number (2) and continue until it is no longer divisible by that prime.
Proceed with the Next Prime: Move to the next smallest prime (3, 5, etc.) and repeat the division.
Continue Until Only Prime Factors Remain: Keep dividing until the final result consists solely of prime numbers.

Example: Find the prime factorization of $\boldsymbol{32760}$ .

$\boldsymbol{32760 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7 \times 13 = 2^3 \times 3^2 \times 5 \times 7 \times 13}$

This is the unique prime factorization for $\boldsymbol{32760}$ .

Applications of the Fundamental Theorem of Arithmetic

Finding the HCF and LCM of Two Numbers:
- Prime factorization is used to determine the HCF and LCM of numbers.
- HCF: Use the lowest powers of the common prime factors.
- LCM: Use the highest powers of all prime factors present in any of the numbers.
Example: Find the HCF and LCM of 6 and 20.
- Prime factorization of 6: $\boldsymbol{2^1 \times 3^1}$
- Prime factorization of 20: $\boldsymbol{2^2 \times 5^1}$
- HCF: $\boldsymbol{2^1 = 2}$
- LCM: $\boldsymbol{2^2 \times 3^1 \times 5^1 = 60}$
Using HCF and LCM Relationship:
- For any two positive integers $\boldsymbol{a}$ and $\boldsymbol{b}$ , the product of their HCF and LCM is equal to the product of the two numbers:
$\boldsymbol{\text{HCF} (a, b) \times \text{LCM} (a, b) = a \times b}$
- This property allows us to calculate the LCM if we already know the HCF.
Divisibility and Factorization Properties:
- The theorem allows us to understand divisibility in numbers. For instance, $\boldsymbol{4^n}$ (where $\boldsymbol{n}$ is a natural number) will never end with zero because its prime factorization does not include the prime number 5.
Simplifying Fractions:
- Using the prime factorization of numerators and denominators, fractions can be simplified by canceling out common prime factors.

Example Problems Using the Fundamental Theorem of Arithmetic

Example: Check if $\boldsymbol{4^n}$ can end with the digit 0.
- If $\boldsymbol{4^n}$ ends with 0, it must be divisible by 5.
- Since $\boldsymbol{4^n = (2^2)^n = 2^{2n}}$ , the only prime factor in $\boldsymbol{4^n}$ is 2, so it can’t end with 0.
Example: Find the HCF and LCM of $\boldsymbol{6}$ , $\boldsymbol{72}$ , and $\boldsymbol{120}$ .
- Prime factorization:
  - $\boldsymbol{6 = 2 \times 3}$
  - $\boldsymbol{72 = 2^3 \times 3^2}$
  - $\boldsymbol{120 = 2^3 \times 3 \times 5}$
- HCF: $\boldsymbol{2^1 \times 3^1 = 6}$
- LCM: $\boldsymbol{2^3 \times 3^2 \times 5^1 = 360}$