‘रचना-प्रयोग’ का मुख्य उद्देश्य क्या है?

व्याकरण का व्यावहारिक प्रयोग सिखाना

‘द्वितीयः’ किस प्रकार की संख्या है?

क्रमवाचक (दूसरा)

State the Fundamental Theorem of Arithmetic.

Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the factors.

Prove that \(\sqrt{2}\) is irrational.

Assume \(\sqrt{2} = p/q\) in lowest terms; squaring gives \(p^2 = 2q^2\), so \(p\) is even, say \(p = 2m\); then \(q^2 = 2m^2\), so \(q\) is also even — contradicting that \(p/q\) is in lowest terms. Hence \(\sqrt{2}\) is irrational.

How do you find the HCF and LCM of two numbers by prime factorisation?

Express each number as a product of primes; the HCF is the product of the smallest powers of common primes, and the LCM is the product of the greatest powers of all primes involved.

What is the relationship between the HCF, LCM and product of two numbers?

For any two positive integers \(a\) and \(b\), \(\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b\).

Find the HCF and LCM of 6 and 20 by prime factorisation.

\(6 = 2 \times 3\) and \(20 = 2^2 \times 5\), so \(\text{HCF} = 2\) and \(\text{LCM} = 2^2 \times 3 \times 5 = 60\) (and \(2 \times 60 = 6 \times 20\)).

Define a rational and an irrational number.

A rational number can be written as \(p/q\) where \(p\) and \(q\) are integers and \(q \ne 0\); an irrational number cannot be written in this form, e.g. \(\sqrt{2}\) and \(\pi\).

Prove that \(\sqrt{3}\) is irrational (outline).

Assume \(\sqrt{3} = p/q\) in lowest terms; then \(p^2 = 3q^2\), so 3 divides \(p\), say \(p = 3m\); then \(q^2 = 3m^2\), so 3 divides \(q\) too — contradicting that \(p/q\) is in lowest terms. Hence \(\sqrt{3}\) is irrational.

When does a rational number have a terminating decimal expansion?

When, in its lowest form \(p/q\), the denominator \(q\) has only 2 and/or 5 as its prime factors \((q = 2^n 5^m)\).

Why can the number \(6^n\) never end with the digit 0 for any natural number \(n\)?

A number ending in 0 must have both 2 and 5 as factors; \(6^n = 2^n 3^n\) has no factor of 5, so it can never end in 0.

State the Fundamental Theorem of Arithmetic.

Every composite number can be expressed as a product of primes, and this factorisation is unique except for the order of the factors.

Real Numbers

Class 10 Mathematics · Chapter 1

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CBSE Class 10 · Mathematics (Standard and Basic)

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CBSE Class 10 · Mathematics (Standard and Basic)

70+ formulas and theorems, all 14 chapters

Marks booster + cheatsheet: key concepts in 2 lines each
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20 marks: Algebra
Board-paper counts, chapter by chapter
12 marks: Trigonometry
11 marks: statistics and probability

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Summary

This chapter builds the foundation of number theory for Class 10. The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a unique product of primes, which lets us find the HCF and LCM of numbers through prime factorisation. A key relationship is that for any two positive integers, \(\text{HCF} \times \text{LCM}\) equals the product of the numbers. The chapter also revisits irrational numbers, proving results such as the irrationality of \(\sqrt{2}\) using the method of contradiction. Together these ideas sharpen the understanding of how integers and primes underpin the rest of arithmetic and algebra.

Fundamental Theorem of ArithmeticHCF and LCM by prime factorisationIrrational numbersProofs of irrationality

Key terms

Fundamental Theorem of Arithmetic

Every composite number has a unique prime factorisation.

HCF

The highest common factor (greatest common divisor) of given numbers.

LCM

The lowest common multiple of given numbers.

Irrational number

A number that cannot be written as a ratio of two integers.

Extra questions & answers

EXERCISE 1.1

EXERCISE 1.2

Try one from this chapter

Which theorem states that every composite number can be expressed as a unique product of primes?

Flashcards & interactive practice