01 · Explore
Introduction to Real Numbers
Real numbers include rational and irrational numbers. This chapter uses prime factors to explain HCF, LCM and why some square roots cannot be fractions.
Think of prime numbers as building blocks for positive integers. For instance, 18 = 2 × 3 × 3. Changing the order of these blocks does not change the number, and no different collection of prime blocks makes 18.
We will use this idea to find the highest common factor and least common multiple of numbers. Then we will use a contradiction: assume a square root is a fraction and show that the assumption cannot work.
The goal is to understand why the rules work. A factorisation explains common factors and multiples; a contradiction proof explains why a square root cannot equal a fraction.
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Question
What is the difference between rational and irrational numbers?
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02 · Explore
The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every composite number can be uniquely factorised into prime numbers.
A composite number is a positive integer greater than 1 that has at least one divisor other than 1 and itself. The Fundamental Theorem of Arithmetic asserts that every such number can be expressed as a product of primes. For example, 253 = 11 × 23.
The theorem further states that this factorisation is unique, meaning there is only one set of prime factors for any given composite number, regardless of the order in which they are written. For instance, 2 × 3 × 5 × 7 is the same as 7 × 5 × 3 × 2.
By convention, we usually write the prime factors in ascending order. For example, the prime factorisation of 32760 is written as 2³ × 3² × 5 × 7 × 13. This standardised order makes the unique factorisation easy to identify.
Prime Factorisation Process
- 1
Composite Number
Start with any composite number x, such as 32760.
- 2
Factor Tree
Break the number down into its smallest prime factors step-by-step.
- 3
Prime Product
Express the number as a product of these primes: 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13.
- 4
Power Notation
Group identical primes using exponents: 2³ × 3² × 5 × 7 × 13.
The sequence of decomposing a composite number into its unique prime factorisation.
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State the Fundamental Theorem of Arithmetic.
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03 · Explore
HCF and LCM by Prime Factorisation
Prime factorisation is a powerful tool for finding the Highest Common Factor (HCF) and Least Common Multiple (LCM) of integers.
The HCF of two or more numbers is the product of the smallest power of each common prime factor involved in the numbers. It represents the largest number that divides all the given numbers without leaving a remainder.
The LCM is the product of the greatest power of each prime factor involved in the numbers. It represents the smallest number that is a multiple of all the given numbers.
For any two positive integers a and b, there is a vital relationship: HCF(a, b) × LCM(a, b) = a × b. This formula allows us to find one value if the other three are known. However, it is important to note that this specific product relationship does not generally hold true for three or more numbers.
Finding HCF and LCM of 6 and 20
6 = 2¹ × 3¹; 20 = 2² × 5¹
Common factor is 2. Smallest power of 2 is 2¹. So, HCF = 2. Prime factors involved are 2, 3, and 5. Greatest powers are 2², 3¹, and 5¹. So, LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60. Check: HCF × LCM = 2 × 60 = 120; Product = 6 × 20 = 120.
Why the two-number rule does not extend directly to three
HCF(12, 18, 30) = 6; LCM(12, 18, 30) = 180
12 = 2² × 3, 18 = 2 × 3², and 30 = 2 × 3 × 5. Use the smallest powers shared by all three for HCF: 2 × 3 = 6. Use the greatest powers present for LCM: 2² × 3² × 5 = 180. Their product is 1080, whereas 12 × 18 × 30 = 6480. The formula HCF × LCM = product is for two positive integers.
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Question
If HCF(306, 657) = 9, find LCM(306, 657).
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04 · Explore
Applications of Prime Factorisation
We can use the uniqueness of prime factorisation to solve theoretical problems regarding number properties.
One interesting application is determining if a number of the form aⁿ can end with a specific digit, such as zero. For a number to end with the digit 0, its prime factorisation must contain both 2 and 5 (since 2 × 5 = 10).
Consider 4ⁿ. The prime factorisation of 4 is 2². Thus, 4ⁿ = (2²)ⁿ = 2²ⁿ. The only prime factor here is 2. Since 5 is not a prime factor, 4ⁿ can never end with the digit 0 for any natural number n.
Similarly, for 6ⁿ, the prime factorisation is (2 × 3)ⁿ = 2ⁿ × 3ⁿ. Since 5 is missing from the factorisation, 6ⁿ also cannot end with the digit 0.
Checking if 6ⁿ ends in 0
6ⁿ = (2 × 3)ⁿ
For a number to end in 0, it must be divisible by 5. The prime factors of 6ⁿ are only 2 and 3. By the Fundamental Theorem of Arithmetic, this factorisation is unique. Since 5 is not a factor, 6ⁿ cannot end in 0.
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Why does 4ⁿ not end with 0?
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