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Real Numbers – Class 10 Mathematics notes, important questions and MCQs

NCERT Class 10 Mathematics, chapter 1. Notes, key terms, important questions with answers and MCQs.

Chapter overview

Every composite number can be written as a product of primes in exactly one way. This lets us find HCF and LCM and prove that numbers like √2 are irrational.

Fundamental Theorem of Arithmetic

Every composite number is a product of primes, unique apart from the order of the factors.
Example: 140 = 2² × 5 × 7.
HCF = product of the smallest powers of common primes; LCM = product of the greatest powers of all primes.

HCF and LCM

For two numbers a and b: HCF(a, b) × LCM(a, b) = a × b.
This product rule does not work for three numbers.
HCF is always a factor of LCM.

Irrational numbers

√p is irrational for every prime p (proved by contradiction).
The sum or product of a non-zero rational and an irrational number is irrational.
Example: 5 − √3 and 2√5 are irrational.

Key terms

Prime factorisation: writing a number as a product of prime numbers
Irrational number: a number that cannot be written as p/q with integers p, q (q ≠ 0)
Co-prime: two numbers whose HCF is 1

Formulas

HCF(a, b) × LCM(a, b) = a × b

Important questions with answers

Find the HCF of 96 and 404 by prime factorisation.2 marks

96 = 2⁵ × 3 and 404 = 2² × 101. Common factor with the lowest power is 2², so HCF = 4.

Prove that √5 is irrational.3 marks

Assume √5 = p/q (co-prime). Then p² = 5q², so 5 divides p; write p = 5r. Then 25r² = 5q², q² = 5r², so 5 divides q. Both divisible by 5 contradicts co-prime, so √5 is irrational.

Prove that √2 is irrational.3 marks

Suppose √2 = a/b with a, b co-prime integers, b ≠ 0. Then 2b² = a², so a² is even and hence a is even; write a = 2c. Then 2b² = 4c², so b² = 2c² and b is even. So 2 divides both a and b, which contradicts that they are co-prime. Hence √2 is irrational.

Find the LCM and HCF of 6, 72 and 120 by prime factorisation, and check whether HCF × LCM equals the product of the three numbers.5 marks

6 = 2 × 3, 72 = 2³ × 3², 120 = 2³ × 3 × 5. HCF = 2 × 3 = 6 (lowest powers of common primes). LCM = 2³ × 3² × 5 = 360 (highest powers of all primes). HCF × LCM = 2160, but 6 × 72 × 120 = 51,840. So the product rule HCF × LCM = product holds only for two numbers, not three.

Quick revision

Prime factorisation of a number is unique.
HCF uses smallest powers; LCM uses greatest powers.
√2, √3, √5 are irrational.
If HCF = 1, the numbers are co-prime and LCM = their product.

Exam tips

In "prove irrational" questions, assume the number is rational, write it as p/q in lowest terms, and reach a contradiction.
Write prime factors with powers (2³ × 3²), not as long products.

Practice MCQs

The prime factorisation of 96 is
1. 2⁵ × 3
2. 2⁴ × 3²
3. 2⁵ × 3²
4. 2³ × 3 × 4
Show answer
Answer: (a) 2⁵ × 3. 96 = 32 × 3 = 2⁵ × 3.
HCF of 12 and 18 is
1. 2
2. 3
3. 6
4. 36
Show answer
Answer: (c) 6. 12 = 2² × 3, 18 = 2 × 3²; smallest powers give 2 × 3 = 6.
Which of these is irrational?
1. √16
2. 0.25
3. 3/4
4. √7
Show answer
Answer: (d) √7. 7 is prime, so √7 is irrational.
If HCF(26, 91) = 13, then LCM(26, 91) is
1. 182
2. 91
3. 2366
4. 364
Show answer
Answer: (a) 182. LCM = 26 × 91 / 13 = 182.
LCM of 6, 72 and 120 is
1. 720
2. 360
3. 1440
4. 240
Show answer
Answer: (b) 360. 6 = 2 × 3, 72 = 2³ × 3², 120 = 2³ × 3 × 5; LCM = 2³ × 3² × 5 = 360.
The product of a non-zero rational number and an irrational number is
1. always irrational
2. always rational
3. an integer
4. zero
Show answer
Answer: (a) always irrational. A non-zero rational times an irrational is irrational.
The smallest number divisible by 1 to 10 is
1. 1260
2. 5040
3. 2520
4. 10!
Show answer
Answer: (c) 2520. LCM(1..10) = 2³ × 3² × 5 × 7 = 2520.
6ⁿ ends with the digit 0 for
1. n = 10
2. n = 5
3. every even n
4. no natural number n
Show answer
Answer: (d) no natural number n. 6ⁿ = 2ⁿ × 3ⁿ has no prime factor 5, so it never ends in 0.

All chapters of Class 10 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.