Real Numbers – CBSE Class 10 Mathematics Notes & Guide

Real numbers form the backbone of many topics in Class 10 mathematics. Understanding how numbers can be broken down into prime factors, how to find the highest common factor (HCF) and the least common multiple (LCM), and mastering divisibility rules will help you solve problems quickly and accurately.

Key concepts

Fundamental Theorem of Arithmetic

Every integer greater than 1 can be expressed uniquely as a product of prime numbers, apart from the order of the factors. This is called the prime factorisation of the number. For example, 84 = 2 × 2 × 3 × 7 = 2²·3·7.

Euclidean algorithm for HCF

The Euclidean algorithm finds the highest common factor of two numbers by repeatedly subtracting the smaller number from the larger (or, more efficiently, using division). If a = b·q + r, then HCF(a,b) = HCF(b,r). The process stops when the remainder becomes zero; the last non‑zero remainder is the HCF.

Least common multiple (LCM)

The LCM of two or more numbers is the smallest positive integer that is divisible by each of them. Using prime factorisation, LCM is obtained by taking the highest power of every prime that appears in any of the numbers.

Prime and composite numbers

• Prime number: a natural number greater than 1 having exactly two distinct divisors – 1 and itself (e.g., 2, 3, 5, 7, 11).
• Composite number: a natural number greater than 1 that has more than two divisors (e.g., 4, 6, 8, 9, 12). The number 1 is neither prime nor composite.

Divisibility rules

Simple tests to check whether a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 11 and 12 without performing full division.

• 2: last digit even.
• 3: sum of digits divisible by 3.
• 4: last two digits form a number divisible by 4.
• 5: last digit 0 or 5.
• 6: divisible by both 2 and 3.
• 8: last three digits form a number divisible by 8.
• 9: sum of digits divisible by 9.
• 10: last digit 0.
• 11: alternating sum of digits is a multiple of 11 (including 0).
• 12: divisible by both 3 and 4.

Important definitions and formulas

• Prime factorisation: n = p₁^(a₁)·p₂^(a₂)·…·pₖ^(aₖ)
• HCF (Euclidean algorithm): HCF(a,b) = HCF(b, a mod b)
• LCM (using prime factors): LCM = ∏ pᵢ^(max(aᵢ,bᵢ,…))
• Relation between HCF and LCM: a·b = HCF(a,b) × LCM(a,b)
• Divisibility rule for 11: |(sum of digits in odd positions) – (sum of digits in even positions)| is a multiple of 11.

Solved examples

Example 1 – Prime factorisation and HCF

Find the HCF of 540 and 864 using the Euclidean algorithm.

1. 864 ÷ 540 = 1 remainder 324 → HCF(540,864) = HCF(540,324)
2. 540 ÷ 324 = 1 remainder 216 → HCF(324,540) = HCF(324,216)
3. 324 ÷ 216 = 1 remainder 108 → HCF(216,324) = HCF(216,108)
4. 216 ÷ 108 = 2 remainder 0 → HCF = 108.

Answer: HCF(540,864) = 108.

Example 2 – LCM by prime factorisation

Find the LCM of 45, 60 and 84.

1. Prime factorise each number:

• 45 = 3²·5
• 60 = 2²·3·5
• 84 = 2²·3·7

1. Take the highest power of each prime appearing:

• 2², 3², 5¹, 7¹

1. LCM = 2²·3²·5·7 = 4·9·5·7 = 1260.

Answer: LCM = 1260.

Example 3 – Using divisibility rules

Check whether 23 456 is divisible by 8 and 11.

• Divisible by 8? Look at the last three digits: 456. 456 ÷ 8 = 57 exactly, so yes.
• Divisible by 11? Alternating sum: (2+4+6) – (3+5) = 12 – 8 = 4. Since 4 is not a multiple of 11, the number is not divisible by 11.

Answer: 23 456 is divisible by 8 but not by 11.

Common mistakes to avoid

• Forgetting that 1 is neither prime nor composite.
• Mixing up the order of subtraction in the Euclidean algorithm; always subtract the smaller from the larger or use the remainder form.
• When finding LCM, taking the lowest power of a prime instead of the highest.
• Applying the divisibility rule for 11 incorrectly – remember to alternate starting from the rightmost digit or be consistent with the chosen direction.
• Assuming the product of HCF and LCM equals the product of the numbers without checking; the relation holds only for two numbers.

Exam tips

• Memorise the short divisibility rules; they save time in MCQs.
• Write down the prime factorisation clearly; it helps avoid errors in HCF/LCM problems.
• For the Euclidean algorithm, keep the remainder column; a small mistake in division can cascade.
• Practice converting word problems into HCF/LCM language (e.g., “find the smallest number that can be divided by 12, 15 and 20”).
• Use the relation a·b = HCF × LCM as a quick check for your answer.

Frequently asked questions

What is the difference between a prime and a composite number?

A prime number has exactly two distinct factors – 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite.

How does the Euclidean algorithm work for three numbers?

First find the HCF of any two numbers, say HCF(a,b). Then find the HCF of this result with the third number: HCF(HCF(a,b),c). The final value is the HCF of all three numbers.

Can the LCM be found without prime factorisation?

Yes. You can use the relation LCM(a,b) = (a×b) ÷ HCF(a,b). For more than two numbers, find the LCM of two numbers first, then use the result with the next number.