Class 10 RD Sharma Solutions- Chapter 1 Real Numbers - Exercise 1.3 Last Updated : 23 Jul, 2025 Comments Improve Suggest changes Like Article Like Report In Chapter 1 of Class 10 Mathematics, RD Sharma explores the topic of "Real Numbers". This chapter delves into various properties of the real numbers including their classification and how they can be expressed as rational and irrational numbers. Exercise 1.3 focuses on understanding and solving the problems related to the Euclidean division algorithm prime factorization and highest common factor (HCF) of the numbers using these methods.Real Numbers The Real numbers include both rational and irrational numbers. Rational numbers can be expressed as fractions or decimals that either terminate or repeat while irrational numbers have non-repeating, non-terminating decimals. Real numbers form the backbone of many mathematical operations and are used to describe quantities along a continuous number line. The properties of real numbers like closure, commutativity, associativity and distributivity play an essential role in simplifying complex mathematical problems.Question 1. Express each of the following integers as a product of its prime.i) 420ii) 468iii) 945iv) 7325Solution:Let us express each of the numbers as a product of prime factors. i) 420Performing prime factorisation of the number, we get,420 = 2 × 2 × 3 × 5 × 7ii) 468Performing prime factorisation of the number, we get,468 = 2 × 2 × 3 × 3 × 13iii) 945Performing prime factorisation of the number, we get,945 = 3 × 3 × 3 × 5 × 7iv) 7325Performing prime factorisation of the number, we get,7325 = 5 × 5 × 293Question 2. Determine the prime factorisation of each of the following positive integer:i) 20570ii) 58500iii) 45470971Solution:Let us express each of the numbers as a product of prime factors. i) 20570Performing prime factorisation of the number, we get,20570 = 2 × 5 × 11 × 11 × 17ii) 58500Performing prime factorisation of the number, we get,58500 = 2 × 2 × 3 × 3 × 5 × 5 × 5 × 13iii) 45470971Performing prime factorisation of the number, we get,45470971 = 7 × 7 × 13 × 13 × 17 × 17 × 19Question 3. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.Solution:Both of these numbers have a common factor of 7. Also, every number is divisible by 1.7 × 11 × 13 + 13 = (77 + 1) × 13 = 78 × 137 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = (7 × 6 × 4 × 3 × 2 + 1) × 5 = 1008 × 5Composite numbers are those numbers which have at least one more factor other than 1.Now,Both of these numbers are even. Therefore, the given two numbers are composite numbersQuestion 4. Check whether 6n can end with the digit 0 for any natural number n.Solution:Since, 6n = (2 × 3)n6n = 2n × 3nAny number can end with 0 if it divisible by 10 or 5 and 2 together. The, prime factorisation of 6n does not contain 5 and 2 as a pair of factors. Therefore, 6n can never end with the digit 0 for any natural number n.Read more :Real NumbersReal-Life Applications of Real NumbersSummary Exercise 1.3 | Set 2 of RD Sharma's Class 10 Mathematics textbook focuses on the properties of rational and irrational numbers. This section covers topics such as identifying rational and irrational numbers, operations on rational and irrational numbers, and proving the irrationality of certain numbers. Students learn to classify numbers as rational or irrational, understand the closure properties of rational and irrational numbers under various operations, and solve problems involving the representation of numbers in decimal form. The exercise also includes questions on finding rational numbers between given numbers and proving statements related to rational and irrational numbers. Comment More infoAdvertise with us M mallikagupta90 Follow Improve Article Tags : Mathematics School Learning Class 10 RD Sharma Solutions RD Sharma Class-10 Maths-Class-10 +2 More Similar Reads Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields 5 min read Basic ArithmeticWhat are Numbers?Numbers are symbols we use to count, measure, and describe things. 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