The Fundamental Theorem of Arithmetic
We have seen that any natural number can be written as a product of its prime factors. For instance,
2 = 2,
4 = 2 × 2,
253 =
Now, let’s flip the question: Can every natural number be created by multiplying prime numbers?.
Pick any natural number, let us say 44, if we factorize it with just primes we get
44=
What about 1771 ?
1771 =
Building Numbers Using Prime Factors
Take any collection of prime numbers, say 2, 3, 7, 11 and 23. If we multiply some or all of these numbers, allowing them to repeat as many times as we wish, we can produce a large collection of positive integers (In fact, infinitely many). Let us list a few :
7 × 11 × 23 =
3 × 7 × 11 × 23 =
2 × 3 × 7 × 11 × 23 =
As you can see, combining primes in different ways creates new numbers. But here’s the big question: If we include all possible prime numbers, can we generate every natural number?
The Infinitude of Primes
There are infinitely many prime numbers, which means if we take all primes and combine them in every possible way, we can generate an infinite collection of numbers. This includes all primes and all possible products of primes.
But what about composite numbers? Can every composite number be expressed as a product of primes? Or could there exist a composite number that cannot be written this way? Let’s explore further.
We are going to use the factor tree with which we are all familiar. Let's take some large number, say, 32760 and factorise it as shown.
Note: You can enter any number of your choice in the input field below.If you are unable to see all the numbers then select the entire factorization tree and move.Do not move the individual element.
Upon factorised 32760 we get:
32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 ×
i.e., 32760 =
Once we have decided that the order will be ascending, then the way the number is factorised, is unique.
Let us try another number, say, 123456789.
This can be written as
We will also have to check if last two factors are primes or not! (Try it out for several other natural numbers using the above factorisation panel).
Upon trying out a variety of large numbers, we can see that every
Theorem 1.1 Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
This theorem is fundamental because it forms the backbone of understanding integers and their properties.
Why is This Important?
Prime factorization helps us:
1.Simplify fractions and find common factors.
2.Understand the nature of numbers in different fields of math.
3.Solve real-world problems like cryptography and computer algorithms.
An equivalent version of the above theorem was (probably) first recorded as "Proposition 14" of Book IX in Euclid’s Elements, before it came to be known as the Fundamental Theorem of Arithmetic.
However, the first correct proof was given by Carl Friedrich Gauss in his Disquisitiones Arithmeticae.
Note: A composite number can be factorised as a
So, for example: we regard factorisation of 210 = 2 × 3 × 5 × 7 is same as writing 210 = 3 × 5 × 7 × 2 (or) any other possible order in which these primes are written.
This fact is also stated in the following form:
The prime factorisation of a natural number is
So now that you’ve seen the power of prime numbers, let’s use this theorem to uncover more mysteries hidden in the world of numbers!
In general, given a composite number x, we factorise it as x =
For example, 32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13
=
Once we have decided that the order will be ascending, then the way the number is factorised, is unique.
Since the theorem is proved, it means that it will always hold
For example, consider the numbers
If the number
That is, the prime factorisation of
So, the uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no other primes in the factorisation of
So, we took the uniqueness part of the theorem and applied it to quickly deduct the point that
Before moving forward, let's revise HCF and LCM.
| S.No | Feature | LCM (Least Common Multiple) | HCF (Highest Common Factor) |
|---|---|---|---|
| 1 | Definition | The smallest number that is a multiple of two or more numbers | The largest number that divides two or more numbers without a remainder |
| 2 | Keywords | Common multiples; Least common; When will they meet again? | Common factors; Greatest common; Dividing evenly |
| 3 | Context | Synchronizing events (timing/schedules); Finding shared cycles or intervals; Combining sets of numbers | Dividing something into equal parts; Sharing items equally; Simplifying fractions or ratios |
| 4 | Usage | Scheduling tasks; Finding the least common time or number; Problem: "When will two bells ring together next?" | Dividing items or objects equally; Simplifying mathematical expressions; Problem: "What is the largest piece that can divide two ropes evenly?" |
| 5 | Example 1 | Find LCM of 4 and 6: Multiples of 4: 4, 8, 12, 16... Multiples of 6: 6, 12, 18... LCM = 12 | Find HCF of 12 and 15: Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 15: 1, 3, 5, 15 HCF = 3 |
| 6 | Example 2 | Two traffic lights blink every 4 minutes and 6 minutes. When will they blink together? Answer: 12 minutes | What is the largest piece of wood that can evenly cut planks of 12 m and 15 m? Answer: 3 m |
| 7 | Formula | Prime factorization: Take the highest powers of all primes | Prime factorization: Take the lowest powers of all common primes |
| 8 | Key Concept | Multiples (focuses on numbers getting bigger) | Factors (focuses on dividing numbers into smaller parts) |
HCF and LCM
We can now see that the prime factorization method actually uses the Fundamental Theorem of Arithmetic to gets its result.
Finding the HCF and LCM of 6 and 20
fill the answers in ascending order only
Prime factors of 6 =
Prime factors of 20 =
i.e. 20 =
So HCF(6,20) =
HCF(6, 20) =
LCM (6, 20) =
In the example above, we can see that HCF(6, 20) × LCM(6, 20) =
But 6 x 20 is also equal to 120.
It can be proven that for any two positive integers a,b:
HCF(a,b) x LCM(a,b) = a x b
This will easily enable us to find the HCF or the LCM if the other is given.
Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.
The prime factorisation of 96 and 404 gives:
96 =
Therefore, the HCF of these two integers is
Now, if you need to find the LCM of 96 and 404 then we use the relationship:
HCF × LCM = Product of the numbers
So, LCM (96, 404) =
Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
We have : 6 = 2 × 3, 72 =
Here,
So, HCF (6, 72, 120) =
So, LCM (6, 72, 120) =
Remark: Notice, 6 × 72 × 120