# Fundamental Theorem of Arithmetic: Statement, Proof and Solved Examples
## Introduction
The Fundamental Theorem of Arithmetic is one of the most important theorems in number theory. It explains how every integer greater than 1 can be represented using prime numbers. The theorem establishes that every such integer is either a prime number itself or can be expressed as a product of prime numbers. Most importantly, this prime factorisation is unique apart from the order in which the prime factors are written.
For example, consider the number 60. It can be expressed as:
60 = 2 × 2 × 3 × 5
Using exponent notation, this becomes:
60 = 2² × 3 × 5
The numbers 2, 3, and 5 are prime numbers. Although 60 can be broken down into factors in several ways, its prime factorisation always contains two factors of 2, one factor of 3, and one factor of 5.
For instance, 60 can first be written as 6 × 10, 4 × 15, or 2 × 30. However, continuing to factorise each composite factor into primes always produces the same collection of prime factors.
This uniqueness is the central idea behind the Fundamental Theorem of Arithmetic.
The theorem is used in simplifying fractions, finding the Highest Common Factor (HCF), calculating the Least Common Multiple (LCM), studying divisibility, and understanding many other results in number theory. It also provides a foundation for more advanced mathematics.
## 1. What Is the Fundamental Theorem of Arithmetic?
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a product of prime numbers, and this representation is unique apart from the order of the prime factors.
In mathematical language, if n is an integer greater than 1, then n can be written as:
n = p₁ × p₂ × p₃ × ... × pₖ
where each pᵢ is a prime number.
The same result can be written using exponents:
n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ
Here, p₁, p₂, and the other p-values are distinct prime numbers, while the exponents are positive integers.
For example:
72 = 2 × 2 × 2 × 3 × 3
Therefore:
72 = 2³ × 3²
The prime numbers 2 and 3 are the only prime factors in this representation. The exponent 3 indicates that 2 appears three times, and the exponent 2 indicates that 3 appears twice.
### Understanding the theorem
The theorem has two main parts.
**Existence:** Every integer greater than 1 can be written as a product of prime numbers. If the integer is already prime, its prime factorisation consists of that number alone.
**Uniqueness:** There is only one prime factorisation when the order of the factors is ignored.
For example, 30 can be written as:
30 = 2 × 3 × 5
Writing it as 5 × 2 × 3 does not produce a different prime factorisation. The factors are simply arranged in a different order.
However, 30 cannot have a different collection of prime factors, such as 2 × 2 × 7, because that product equals 28 rather than 30.
The uniqueness condition is what makes the theorem fundamental to arithmetic.
## 2. Important Definitions
Before studying the proof, it is helpful to understand the mathematical terms used in the theorem.
### Prime number
A prime number is an integer greater than 1 that has exactly two positive divisors: 1 and itself.
Examples include:
2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
The number 2 is the only even prime number. Every other even integer greater than 2 is divisible by 2 and therefore is not prime.
### Composite number
A composite number is an integer greater than 1 that has more than two positive divisors.
For example, 12 is composite because it is divisible by 1, 2, 3, 4, 6, and 12.
A composite number can be broken down into smaller factors until all the factors are prime.
### Factor
A factor of an integer is a number that divides it exactly, leaving a remainder of zero.
For example, 4 is a factor of 20 because 20 ÷ 4 = 5.
### Prime factorisation
Prime factorisation means expressing an integer as a product of prime numbers.
For example:
84 = 2 × 2 × 3 × 7
In exponent form:
84 = 2² × 3 × 7
### Exponent
An exponent tells us how many times a base appears as a repeated factor.
For example:
5³ = 5 × 5 × 5 = 125
Therefore, the expression 2³ × 3² means:
2 × 2 × 2 × 3 × 3.
These definitions are important because the theorem depends on the distinction between general factors and prime factors.
## 3. Statement of the Fundamental Theorem of Arithmetic
The formal statement is:
**Every integer greater than 1 is either a prime number or can be expressed as a product of prime numbers, and this representation is unique apart from the order of the factors.**
Let n be an integer greater than 1.
Then:
n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ
where:
- Each pᵢ is a distinct prime number.
- Each exponent aᵢ is a positive integer.
- The product equals n.
- The prime factorisation is unique apart from rearranging the factors.
Consider 100.
Its prime factorisation is:
100 = 2 × 2 × 5 × 5
Therefore:
100 = 2² × 5²
We can also write 100 as 4 × 25, 10 × 10, or 2 × 50. These are valid factorisations, but they are not fully broken down into primes.
After further factorisation, each expression produces the same prime factors: two 2s and two 5s.
This illustrates the statement of the theorem.
## 4. Proof of the Fundamental Theorem of Arithmetic
A complete proof requires establishing both existence and uniqueness. The existence part shows that every integer greater than 1 has a prime factorisation. The uniqueness part shows that no integer greater than 1 has two genuinely different prime factorisations.
### Part A: Proof of existence
Let n be an integer greater than 1.
There are two possibilities.
**Case 1: n is prime.**
If n is prime, it is already a product consisting of one prime number.
Therefore, its prime factorisation exists.
**Case 2: n is composite.**
If n is composite, it can be written as:
n = a × b
where a and b are integers greater than 1 and smaller than n.
If a and b are both prime, then n is already expressed as a product of prime numbers.
If either factor is composite, that factor can be broken down further into smaller positive integers greater than 1.
For example:
60 = 6 × 10
Both 6 and 10 are composite, so we continue:
6 = 2 × 3
10 = 2 × 5
Substituting gives:
60 = 2 × 3 × 2 × 5
Rearranging the factors:
60 = 2 × 2 × 3 × 5
The process must eventually stop because every factorisation of a composite integer greater than 1 produces factors smaller than the original integer. We cannot keep replacing an integer with smaller integers greater than 1 indefinitely.
Eventually, all the factors are prime.
Therefore, every integer greater than 1 can be expressed as a product of prime numbers.
This proves the existence of prime factorisation.
### Part B: Proof of uniqueness
The uniqueness proof relies on an important result known as Euclid's lemma:
**If a prime number p divides the product ab, then p divides a or p divides b.**
This result can be extended to a product containing any finite number of factors.
Suppose an integer n greater than 1 has two prime factorisations:
n = p₁ × p₂ × ... × pₘ
and:
n = q₁ × q₂ × ... × qₖ
Here, all the p-values and q-values are prime numbers. Repeated primes are allowed within either product.
Because both expressions equal n, we have:
p₁ × p₂ × ... × pₘ = q₁ × q₂ × ... × qₖ
The prime p₁ divides the left-hand side, so it divides the right-hand product as well.
By Euclid's lemma, p₁ must divide at least one of the prime numbers q₁, q₂, ..., qₖ.
Since q₁, q₂, and the other q-values are prime, a prime number can divide one of them only if it is equal to that prime number.
Therefore, p₁ equals one of the q-values.
We can rearrange the q-factors so that this matching prime appears first. Both sides now contain the same prime factor, which can be cancelled:
p₂ × ... × pₘ = q₂ × ... × qₖ
Repeating this argument matches each prime factor on one side with an equal prime factor on the other side.
No unmatched prime factor can remain on either side, because the product of integers greater than 1 cannot equal 1.
Consequently, the two products contain exactly the same prime factors with exactly the same multiplicities. They may differ in order, but they do not differ in their prime composition.
Therefore, the prime factorisation of n is unique apart from the order of the factors.
This proves the Fundamental Theorem of Arithmetic.
## 5. Prime Factorisation Methods
There are several practical methods for finding the prime factorisation of an integer. Two commonly used methods are repeated division and factor trees.
### Method 1: Repeated division
In this method, divide the number by the smallest possible prime factor and continue dividing the resulting quotient until the quotient becomes 1.
Consider the number 120.
First, divide by 2:
120 ÷ 2 = 60
Divide again by 2:
60 ÷ 2 = 30
Continue:
30 ÷ 2 = 15
Now 15 is not divisible by 2. Divide it by 3:
15 ÷ 3 = 5
Finally:
5 ÷ 5 = 1
The prime factors are:
2, 2, 2, 3, 5
Therefore:
120 = 2³ × 3 × 5
### Method 2: Factor tree
A factor tree breaks a composite number into two factors repeatedly until all branches end in prime numbers.
Consider 84.
Start with:
84 = 12 × 7
The number 7 is prime, but 12 is composite.
Break down 12:
12 = 3 × 4
Break down 4:
4 = 2 × 2
The final prime factors are:
2, 2, 3, 7
Therefore:
84 = 2² × 3 × 7
A factor tree is useful for visual learners because it shows how the original number is broken down step by step.
### Which method is better?
Repeated division is usually convenient for larger numbers and systematic calculations. Factor trees are helpful for understanding the structure of a factorisation.
Both methods produce the same prime factors because of the uniqueness guaranteed by the theorem.
## 6. Solved Examples of the Fundamental Theorem of Arithmetic
### Example 1: Find the prime factorisation of 36
Divide 36 by the smallest possible prime number.
36 = 2 × 18
18 = 2 × 9
9 = 3 × 3
Therefore:
36 = 2 × 2 × 3 × 3
In exponent form:
**36 = 2² × 3²**
### Example 2: Find the prime factorisation of 72
Start with 72.
72 = 2 × 36
36 = 2 × 18
18 = 2 × 9
9 = 3 × 3
Therefore:
72 = 2 × 2 × 2 × 3 × 3
So:
**72 = 2³ × 3²**
### Example 3: Find the prime factorisation of 150
Divide by 2:
150 = 2 × 75
Divide 75 by 3:
75 = 3 × 25
Break down 25:
25 = 5 × 5
Therefore:
150 = 2 × 3 × 5 × 5
So:
**150 = 2 × 3 × 5²**
### Example 4: Find the prime factorisation of 210
210 = 2 × 105
105 = 3 × 35
35 = 5 × 7
All the factors are prime, so:
**210 = 2 × 3 × 5 × 7**
### Example 5: Find the prime factorisation of 360
360 = 2 × 180
180 = 2 × 90
90 = 2 × 30
30 = 2 × 15
15 = 3 × 5
Therefore:
360 = 2 × 2 × 2 × 2 × 3 × 5
In exponent notation:
**360 = 2⁴ × 3² × 5**
### Example 6: Show that the prime factorisation of 96 is unique
First, find its prime factorisation:
96 = 2 × 48
48 = 2 × 24
24 = 2 × 12
12 = 2 × 6
6 = 2 × 3
Therefore:
96 = 2⁵ × 3
Suppose another method begins with 96 = 12 × 8. The prime factorisation of 12 is 2² × 3, and that of 8 is 2³.
Combining them gives:
96 = 2² × 3 × 2³
Therefore:
96 = 2⁵ × 3
Both methods produce the same prime factors with the same multiplicities. The uniqueness part of the theorem guarantees that every correct complete prime factorisation of 96 must have this form, apart from the order of the factors.
## 7. Applications of the Fundamental Theorem of Arithmetic
### Finding the HCF
The Highest Common Factor is the greatest positive integer that divides two or more integers exactly.
Prime factorisation makes it possible to identify the common prime factors of two numbers.
Consider 48 and 72.
Prime factorisation of 48:
48 = 2⁴ × 3
Prime factorisation of 72:
72 = 2³ × 3²
The common prime factors are 2 and 3. To find the HCF, take each common prime with the smaller exponent.
HCF = 2³ × 3
Therefore:
**HCF(48, 72) = 24**
### Finding the LCM
The Least Common Multiple is the smallest positive integer that is a multiple of both given integers.
Using the same numbers:
48 = 2⁴ × 3
72 = 2³ × 3²
For the LCM, take each prime factor with the greatest exponent appearing in either factorisation.
LCM = 2⁴ × 3²
LCM = 16 × 9
**LCM(48, 72) = 144**
The HCF uses the smaller powers of common primes, whereas the LCM uses the largest powers appearing in either number.
### Simplifying fractions
The theorem supports fraction simplification by identifying common prime factors in the numerator and denominator.
Consider:
84/126
Prime factorisation gives:
84 = 2² × 3 × 7
126 = 2 × 3² × 7
The common factors are 2 × 3 × 7 = 42.
Dividing both terms by 42:
84/126 = 2/3
Therefore, the simplified fraction is 2/3.
### Checking perfect squares
A positive integer is a perfect square if every prime factor in its prime factorisation has an even exponent.
For example:
144 = 2⁴ × 3²
Both exponents are even, so 144 is a perfect square:
144 = 12²
Consider 72:
72 = 2³ × 3²
The exponent of 2 is odd. Therefore, 72 is not a perfect square.
This criterion follows from the fact that squaring an integer doubles the exponents in its prime factorisation.
### Checking perfect cubes
Similarly, a positive integer is a perfect cube if every prime factor has an exponent divisible by 3.
For example:
216 = 2³ × 3³
Therefore:
216 = 6³
By contrast, 72 = 2³ × 3² is not a perfect cube because the exponent of 3 is not divisible by 3.
### Understanding divisibility
Prime factorisation can help determine whether one positive integer divides another.
For example:
12 = 2² × 3
60 = 2² × 3 × 5
All the prime factors of 12 occur in 60 with at least the same exponents. Therefore, 12 divides 60 exactly.
The same principle can be applied to larger integers by comparing their prime exponents.
## 8. Common Mistakes
### Mistake 1: Treating 1 as a prime number
The number 1 is not prime because it has only one positive divisor. Prime numbers have exactly two positive divisors.
The theorem therefore applies to integers greater than 1, not to 1.
### Mistake 2: Stopping at a composite factor
For example, 60 = 6 × 10 is a valid factorisation, but it is not a prime factorisation because 6 and 10 are composite.
Continue until all factors are prime:
60 = 2² × 3 × 5
### Mistake 3: Forgetting repeated prime factors
For example:
36 = 2 × 2 × 3 × 3
Writing only 2 × 3 would produce 6, not 36.
Every occurrence of each prime factor must be included.
### Mistake 4: Assuming order changes uniqueness
The products 2 × 3 × 5 and 5 × 2 × 3 are the same prime factorisation. The theorem says that the prime factors and their multiplicities are unique, not that their order is fixed.
### Mistake 5: Confusing HCF and LCM
For prime factorisations, the HCF uses the smallest exponents of common prime factors. The LCM uses the largest exponent of every prime factor present in either number.
Using the wrong exponents produces an incorrect result.
### Mistake 6: Ignoring the prime definition
A factor is not necessarily prime. For instance, 15 is a factor of 60, but it is composite because 15 = 3 × 5.
Always verify that the final factors are prime.
## 9. Practice Questions
Try these problems to test your understanding.
1. State the Fundamental Theorem of Arithmetic.
2. Find the prime factorisation of 48.
3. Find the prime factorisation of 180.
4. Express 252 as a product of prime factors in exponent form.
5. Find the HCF of 36 and 60 using prime factorisation.
6. Find the LCM of 24 and 90 using prime factorisation.
7. Determine whether 225 is a perfect square using prime factorisation.
8. Determine whether 216 is a perfect cube.
9. Explain why the number 1 is not prime.
10. Explain the uniqueness part of the Fundamental Theorem of Arithmetic.
### Answers
**1.** Every integer greater than 1 is prime or can be written as a product of prime numbers, uniquely apart from the order of the factors.
**2.** 48 = 2⁴ × 3.
**3.** 180 = 2² × 3² × 5.
**4.** 252 = 2² × 3² × 7.
**5.** 36 = 2² × 3² and 60 = 2² × 3 × 5. HCF = 2² × 3 = 12.
**6.** 24 = 2³ × 3 and 90 = 2 × 3² × 5. LCM = 2³ × 3² × 5 = 360.
**7.** 225 = 3² × 5². All exponents are even, so 225 is a perfect square: 225 = 15².
**8.** 216 = 2³ × 3³. Both exponents are multiples of 3, so 216 is a perfect cube: 216 = 6³.
**9.** The number 1 has only one positive divisor, whereas a prime number must have exactly two positive divisors.
**10.** Uniqueness means that every complete prime factorisation of an integer greater than 1 contains exactly the same prime factors with the same multiplicities, regardless of their order.
## 10. Frequently Asked Questions
### What is the Fundamental Theorem of Arithmetic?
It states that every integer greater than 1 can be expressed as a product of prime numbers in a way that is unique apart from the order of the factors.
### Who developed the Fundamental Theorem of Arithmetic?
The theorem belongs to the mathematical tradition of number theory associated with Euclid and later mathematicians. The *Elements* of Euclid contains foundational results about primes and divisibility. The modern theorem is a central result of elementary number theory.
### Why is the theorem called fundamental?
It provides the foundation for understanding the prime structure of integers. Many results involving divisibility, common factors, fraction simplification, and integer factorisation depend on this property.
### Is the prime factorisation of a number always unique?
Yes, for every integer greater than 1, the prime factorisation is unique apart from the order of the factors.
### Is 1 included in the theorem?
The theorem applies to integers greater than 1. The number 1 is excluded because it is not prime and does not have a prime factorisation in the usual sense.
### What is the difference between factorisation and prime factorisation?
Factorisation expresses a number as a product of factors. Prime factorisation expresses it as a product in which every factor is prime.
For example, 60 = 6 × 10 is a factorisation, while 60 = 2² × 3 × 5 is its prime factorisation.
### How is the theorem used to calculate HCF?
Write each number as a product of primes and multiply the common prime factors using their smallest exponents.
### How is the theorem used to calculate LCM?
Write each number as a product of primes and multiply all prime factors that appear, using the largest exponent for each.
### What is the prime factorisation of 100?
100 = 2 × 2 × 5 × 5 = 2² × 5².
### Why is prime factorisation useful in advanced mathematics?
Prime factorisation helps analyse divisibility and integer structure. It is important in number theory and supports algorithms involving greatest common divisors and other arithmetic operations.
## Conclusion
The Fundamental Theorem of Arithmetic is a central theorem of number theory. It states that every integer greater than 1 is either prime or can be expressed as a product of prime numbers, with a unique prime factorisation apart from the order of the factors.
The existence proof shows that repeated factorisation eventually produces prime numbers. The uniqueness proof relies on the property that if a prime divides a product, it must divide at least one of the factors.
The theorem has practical applications in finding HCF and LCM, simplifying fractions, checking perfect squares and cubes, and analysing divisibility. Learning to use repeated division and factor trees makes prime factorisation easier and provides a foundation for more advanced mathematical concepts.
Understanding this theorem helps students recognise the fundamental role of prime numbers in arithmetic and develop stronger problem-solving skills in mathematics.