Exercise 1.3 Real Numbers | CBSE Class 10th Mathematics | Solutions

Question 1. Prove that √5 is irrational.

Solution : Let us assume, to the contrary, that √5 is rational.
That is, we can find integers a and b (≠ 0) such that :

		a
√5	=
		b

⋅

Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.
So, b √5 = a⋅

Real Numbers - Exercise 1.2 | CBSE Class 10th Mathematics | Solutions

Question 1. Express each number as a product of its prime factors:
(i) 140   (ii) 156    (iii) 3825   (iv) 5005    (v) 7429

Solution (i) To express the given number 140 as product of its prime factors, we employ the division method as shown below :

2	|	140
2	|	70
5	|	35
7	|	7
	|	1

∴ 140 = 2×2×5×7
       = 2²×5×7

Real Numbers - Exercise 1.1 | CBSE Class 10th Mathematics | Solutions

Points to remember

• An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. The word algorithm comes from the name of the 9th century Persian mathematician al-Khwarizmi.
• A lemma is a proven statement used for proving another statement.
• Euclid’s Division Lemma : Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
• Euclid’s division algorithm : Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers c and d, with c > d, is obtained as follows: