Real Numbers- Euclid’s Division Lemma
* Euclid’s division lemma states that for any two positive integers ‘a’ and ‘b’ we can find two whole numbers ‘q’ and ‘r’ such that: a= bq + r, where 0< r < b.
* Euclid’s division lemma can be used to find the highest common factor of any two positive integers.
* Euclid’s division lemma can be used to show common properties of numbers.
Real Numbers- Fundamental Theorem of Arithmetic
* Fundamental Theorem of Arithmetic states that every composite number can be expressed or factorised as a product of prime numbers and this factorisation is unique except in the order of the prime factors.
* HCF of two numbers = The products of Terms containing smallest powers of common Prime Factors of the two numbers.
* LCM of two numbers = The products of Terms containing greatest powers of all Prime Factors of the two numbers.
* Product of two positive integers ‘a’ and ‘b’ is equal to the product of the HCF (a,b) and the LCM (a.b)
Real Numbers – Revisiting Rational And Irrational Numbers
* If p/q is a rational number, such that the prime factorisation of q is of the form 2^{a} 5^{b} , where a and b are positive integers, then the decimal expansion of the rational number p/q terminates.
* If p/q is a rational number such that the prime factorisation of q is not of the form 2^{a} 5^{b} , where a and b are positive integers, then the decimal expansion of the rational number p/q terminate and is recurring.