Facts that Matter- Real Numbers - Class 10 PDF Download

 Facts that Matter

• Algorithm
  An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
• Lemma
  A lemma is a proven statement used for proving another statement.
• Euclid’s Division Lemma
  For any two given positive integers ‘a’ and ‘b’ there exists unique whole numbers ‘q’ and ‘r’ such that
  a = bq + r, where 0 ≥ r < b
  Here, a = Dividend, b = Divisor
  q = Quotient, r = Remainder
  i.e., Dividend = (Divisor × Quotient) + Remainder
• Euclid’s Division Algorithm
  Euclid’s Division Algorithm is a technique to compute the HCF of two positive integers ‘a’ and ‘b’ (a > b) using the following steps:
  Step-I: Applying Euclid’s Lemma to a and b to find whole numbers ‘q’ and ‘r’ such that:
  a = bq + r,  0 ≥ r < b  
  Step-II: If r = 0 then ‘b’ is the HCF of ‘a’ and ‘b’. If r ≠ 0 then apply the Euclid’s division lemma to ‘b’ and ‘r’.
  Step-III: Continue the process till the remainder is zero, i.e., repeat the step-II again and again until r = 0. Then, the divisor at this stage will be the required H.C.F.
  

Note: 
I. We state Euclid’s Divison Algorithm for positive integers only but it can be extended for all integers except zero i.e., b ≠ 0.
II. When ‘a’ and ‘b’ are two positive integers such that a = bq + r, 0 ≤ r < b then HCF (a, b) = HCF (b, r).

Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.