Every whole number above 1 breaks into primes in exactly one way. This course builds on that fact to find HCF and LCM, and then uses it to prove that some numbers can never be written as fractions.
The course follows the NCERT Mathematics chapter Real Numbers and has 4 lessons. Each lesson explains the ideas with worked examples and ends with practice questions and full answers. The numbers in every example and question are our own, so the lessons can be used alongside the textbook exercises.
- The Fundamental Theorem of Arithmetic, Every composite number is a product of primes, and in only one way. What the word 'uniquely' is doing in that sentence, how to write a number in standard prime-power form, and why 1 is excluded.
- HCF and LCM from Prime Factorisation, Read the HCF and LCM straight off two prime factorisations: smaller exponents for one, larger for the other. Includes the HCF x LCM identity, why it holds, and why it fails for three numbers.
- Proving a Number Is Irrational, The proof by contradiction that root 2 is irrational, written as a shape you can reuse for root 3, root 5 and root 7, plus where the proof correctly breaks down for root 4.
- Real Numbers: Mixed Practice, Twelve questions across the whole chapter, factorisation, HCF and LCM, and irrationality, with full answers. Several deliberately restrict the method to test reasoning rather than computation.
By the end you will be able to:
- State the Fundamental Theorem of Arithmetic and prime-factorise any number.
- Find the HCF and LCM of two or three numbers from their prime factorisations, and use HCF × LCM = product.
- Prove that numbers such as the square root of 2, 3 or 5 are irrational.
- Decide whether sums and products involving irrational numbers are rational or irrational.