Real Numbers · Lesson 3 of 5
Fundamental Theorem of Arithmetic
“Proof that no matter how big or chaotic a number looks, it can always be broken down into its original, unique recipe of primes.”
• Understand prime factorisation. • State the Fundamental Theorem of Arithmetic. • Factor any composite number into primes. • Know why the factorisation is unique. • Solve textbook questions based on prime factorisation. • Understand why this theorem is so important in mathematics.
Introduction
Can every number be broken into smaller building blocks?
Can we break 2 further?
No.
Can we break 3 further?
No.
So, 2 and 3 are the basic building blocks of 12.
Prime numbers are the basic building blocks of composite numbers. A composite number can be formed by multiplying prime numbers.
Prime and Composite Numbers
A natural number greater than 1 that has exactly two factors, 1 and the number itself, is called a prime number.
A natural number greater than 1 that has more than two factors is called a composite number.
| Prime Numbers | Composite Numbers |
|---|---|
| 2 | 4 |
| 3 | 6 |
| 5 | 8 |
| 7 | 9 |
| 11 | 10 |
The number 1 is neither prime nor composite because it has only one factor.
What Is Prime Factorisation?
Expressing a composite number as a product of prime numbers is called prime factorisation.
To find the prime factorisation of a number, we repeatedly divide or break the number into smaller factors. We stop only when every remaining factor is a prime number.
Problem
Express 72 as a product of prime factors.
- 1.Begin by splitting 72 into two factors.
- 2.72 = 2 × 36
- 3.Now factorise 36: 36 = 2 × 18.
- 4.Factorise 18: 18 = 2 × 9.
- 5.Factorise 9: 9 = 3 × 3.
- 6.All the remaining factors are prime numbers.
- 7.Therefore, 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².
Prime factorisation can be found using a factor tree or repeated division. Both methods produce the same prime factors.
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of prime numbers, and this factorisation is unique except for the order in which the prime factors are written.
The theorem makes two important statements. First, every composite number can be broken into prime factors. Second, the prime factorisation of a number is unique.
What Does Unique Factorisation Mean?
Unique does not mean that the prime factors must be written in one fixed order. It means that no different collection of prime factors can produce the same number.
Writing 60 = 6 × 10 is not its prime factorisation because 6 and 10 are composite numbers. Continue until every factor is prime: 60 = 2² × 3 × 5.
Why Is This Theorem Important?
The Fundamental Theorem of Arithmetic gives every composite number a unique prime identity. Once a number is written as a product of primes, many questions about its factors, divisibility, HCF and LCM become easier to solve.
- It allows us to find the HCF and LCM of numbers.
- It helps us study divisibility and factors.
- It is used to determine whether certain numerical forms are possible.
- It forms an important foundation of number theory.
- Prime factorisation is also used in modern computer security and cryptography.
Calculation of HCF and LCM Using Prime Factorisation
One of the most useful applications of prime factorisation is calculating the Highest Common Factor and the Lowest Common Multiple of two or more numbers.
The HCF of two or more numbers is the greatest number that divides each of them exactly.
The LCM of two or more numbers is the smallest positive number that is a multiple of each of them.
Write the prime factorisation of every number. Select only the prime factors common to all the numbers and take the smallest power of each common prime factor.
Write the prime factorisation of every number. Select every prime factor appearing in any of the numbers and take the greatest power of each prime factor.
HCF → Common prime factors → Smallest powers LCM → All prime factors → Greatest powers
HCF and LCM of Two Numbers
Problem
Find the HCF and LCM of 36 and 60 using prime factorisation.
- 1.Write the prime factorisation of both numbers.
- 2.36 = 2² × 3²
- 3.60 = 2² × 3 × 5
- 4.For the HCF, select the common prime factors 2 and 3 with their smallest powers.
- 5.HCF = 2² × 3 = 12.
- 6.For the LCM, select all prime factors with their greatest powers.
- 7.LCM = 2² × 3² × 5 = 180.
HCF and LCM of Three Numbers
Problem
Find the HCF and LCM of 12, 18 and 30 using prime factorisation.
- 1.Write the prime factorisation of all three numbers.
- 2.12 = 2² × 3
- 3.18 = 2 × 3²
- 4.30 = 2 × 3 × 5
- 5.The prime factors common to all three numbers are 2 and 3.
- 6.Using their smallest powers, HCF = 2 × 3 = 6.
- 7.For the LCM, take all prime factors with their greatest powers.
- 8.LCM = 2² × 3² × 5 = 180.
For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b This relationship can be used to verify an answer.
Real-Life Applications of HCF and LCM
HCF and LCM are useful when quantities must be divided into the largest equal groups or when repeating events must occur together again.
Use HCF when objects or measurements must be divided into the largest possible equal groups or equal-sized pieces without leaving anything unused.
Use LCM when two or more events repeat after fixed intervals and we need to find when they will occur together again.
Problem
There is a circular path around a sports field. Messi takes 18 minutes to complete one round of the field, while Ronaldo takes 12 minutes for the same. They both start at the same point, at the same time, and move in the same direction. After how many minutes will they meet again at the starting point?
- 1.Messi returns to the starting point after every 18 minutes.
- 2.Ronaldo returns to the starting point after every 12 minutes.
- 3.To find when they will be at the starting point together again, calculate the LCM of 18 and 12.
- 4.18 = 2 × 3²
- 5.12 = 2² × 3
- 6.LCM = 2² × 3² = 36.
- 7.Therefore, Messi and Ronaldo will meet again at the starting point after 36 minutes.
Words such as together again, repeat together, meet again or occur at the same time usually indicate that LCM should be used.
Application of Prime Factorisation
Problem
Check whether 6ⁿ can end with the digit 0 for any natural number n.
- 1.A number that ends with the digit 0 must be divisible by 10.
- 2.The prime factorisation of 10 is 2 × 5.
- 3.Therefore, a number ending in 0 must contain both 2 and 5 as prime factors.
- 4.Now, 6 = 2 × 3.
- 5.Therefore, 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ.
- 6.The prime factorisation of 6ⁿ contains only the prime factors 2 and 3.
- 7.It does not contain the prime factor 5.
- 8.Therefore, 6ⁿ cannot be divisible by 10.
- 9.Hence, 6ⁿ cannot end with the digit 0 for any natural number n.
A number can end in 0 only when its prime factorisation contains at least one pair of 2 and 5. Since 6ⁿ contains no factor 5, it can never end in 0.
• Stopping the factorisation while composite factors still remain. • Treating 1 as a prime number. • Assuming that changing the order of prime factors creates a different factorisation. • Using the greatest powers while finding the HCF. • Using only common prime factors while finding the LCM. • Forgetting that a number ending in 0 must contain both 2 and 5 as prime factors. • Using HCF instead of LCM in a problem involving repeating events.
Quiz
Which of the following is the prime factorisation of 72?
While finding the HCF using prime factorisation, which powers are selected?
What is the LCM of 12 and 18?
Why can 6ⁿ never end with the digit 0?
Two events repeat every 12 minutes and 18 minutes. After how many minutes will they occur together again?
Practice Problems
- Express 84 as a product of prime factors.
- Express 156 as a product of prime factors.
- Find the HCF and LCM of 24 and 36 using prime factorisation.
- Find the HCF and LCM of 45 and 75 using prime factorisation.
- Find the HCF and LCM of 18, 24 and 30.
- Find the HCF and LCM of 28, 42 and 56.
- Two bells ring after every 20 minutes and 30 minutes. If they ring together now, after how many minutes will they ring together again?
- Three runners complete one round of a track in 8 minutes, 12 minutes and 18 minutes. After how many minutes will all three return to the starting point together?
- Check whether 12ⁿ can end with the digit 0 for any natural number n.
- Explain why 2 × 3 × 5 × 7 + 1 is not divisible by any of 2, 3, 5 or 7.
Key Takeaways
• Prime numbers have exactly two factors, while composite numbers have more than two factors. • Prime factorisation expresses a composite number as a product of prime numbers. • Every composite number has a unique prime factorisation, except for the order of its factors. • For the HCF, select common prime factors with their smallest powers. • For the LCM, select all prime factors with their greatest powers. • HCF is useful when dividing quantities into the largest equal groups. • LCM is useful when finding when repeating events will occur together again. • A number ending in 0 must contain both 2 and 5 in its prime factorisation.