R.D. Sharma Mathematics X Chapter 1

Study Guide: Chapter 1: Real Numbers Divisibility 

A non-zero integer ‘a’ is said to divide an integer ‘b’ if there exists an integer ‘c’ such that b=ac. o We write a | b to show that ‘a’ divides ‘b’. If ‘b’ is not divisible by ‘a’, we write a ∤ b. o ‘b’ is known as the dividend, ‘a’ is known as the divisor, and ‘c’ is known as the quotient. Ex: 4 | 20 because: Here, a = 4 (divisor), b = 20 (dividend), c = 5 (quotient).

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Properties of divisibility: i. ±1|a ii. a|0 iii. 0 does not divide any integer iv. If ‘a’ is a non-zero integer, and ‘b’ is any integer, then: a | b => a | -b, -a | b, and – a | -b v. If ‘a’ and ‘b’ are non-zero integers, then: a | b and b | a => a = ±b vi. If ‘a’ is a non-zero integer and ‘b’ and ‘c’ are any two integers, then: | | | | { (for any integer ‘x’) | vii. If ‘a’ and ‘c’ are non-zero integers and ‘b’ and ‘d’ are any two integers, then a. a | b and c | d => ac | bd b. ac | bc => a | b

Euclid’s Division Lemma  

A lemma is a proven statement which is used to prove other statements. Euclid’s Division Lemma: Let ‘a’ and ‘b’ be any two positive integers. Then, there exist unique integers ‘q’ and ‘r’ such that: a = bq + r, 0 ≤ r < b. (r = 0 if b | a) o Note: This is nothing but a restatement of the long division process that we do; q is the quotient and r is the remainder. o This lemma holds true for negative integers if: 0 ≤ r < |b|

Euclid’s Division Algorithm  

An algorithm is a series of well defined steps which provide a procedure of calculation, repeated successively on the results of earlier steps till the desired result is obtained. If an integer ‘c’ divides each one of several integers x1, x2, … xn, then it is called a common divisor of these integers. o The largest or greatest among the common divisors of two or more integers is called the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) or Greatest Common Factor (GCF) of the given integers. o Theorem: If ‘a’ and ‘b’ are positive integers such that a = bq + r, then every common divisor of ‘a’ and ‘b’ is a common divisor of ‘b’ and ‘r’, and vice-versa.