Real Numbers
CBSE Class 10 · Mathematics · Notes, formulas and practice questions
This chapter turns on one key idea: every integer greater than 1 factorises uniquely into primes, and this lets us find HCF and LCM, relate them, and prove that radicals like √2, √3 and √5 are irrational by contradiction.
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The chapter opens with the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be written as a product of primes in exactly one way, apart from their order. This single fact gives a systematic method for working with factors and multiples. Instead of listing all divisors or testing divisibility, we factor each number into primes and record their exponents. The theorem also underpins the irrationality proofs later in the chapter.
Once numbers are in prime-factor form, HCF and LCM fall out by comparing exponents. The HCF keeps each common prime with the smallest exponent present; the LCM takes every prime that appears anywhere with the largest exponent. For a pair of numbers, a useful identity connects them: HCF × LCM equals the product of the two numbers. A frequent error is applying this identity to three numbers, where no such simple relation exists.
The second half deals with irrational numbers. A rational number can always be written as a fraction p/q in lowest terms. To show √2, √3 or √5 is irrational, assume it is rational and write it as p/q; squaring leads to a common factor emerging between p and q, contradicting the assumption of lowest terms. The pattern is identical for each, only the prime inside the radical changes.
The Fundamental Theorem also answers questions about what a number can look like. If a natural number ends in the digit 0, its prime factorisation must contain both 2 and 5, because it is divisible by 10 = 2 × 5. Applying this test to numbers like 4ⁿ shows they can never end in 0 because 5 is absent. These applications connect abstract prime factorisation to observable properties of numbers.
Key terms
- Fundamental Theorem of Arithmetic
- Every positive integer greater than 1 is either prime or can be expressed as a product of primes. That product is unique except for the order of factors. For example, 90 = 2 × 3² × 5, and no other set of primes gives 90.
- Prime factorisation
- The process of writing a number as a product of its prime factors, usually with like primes collected into powers. For instance, 84 = 2² × 3 × 7. This form is the starting point for finding HCF and LCM.
- Highest Common Factor (HCF)
- The largest positive integer that divides each of the given numbers. Using prime factorisation, take each prime that appears in every number, raise it to the smallest exponent found, and multiply. For example, HCF of 18 and 24 is 2 × 3 = 6.
- Lowest Common Multiple (LCM)
- The smallest positive integer that is a multiple of each of the given numbers. Using prime factorisation, take every prime that appears in any number, raise it to the greatest exponent found, and multiply. For example, LCM of 18 and 24 is 2³ × 3² = 72.
- Irrational number
- A real number that cannot be written as a fraction p/q with p and q integers and q ≠ 0. Its decimal expansion never terminates and never repeats. Common examples include √2, √3 and √5.
- Proof by contradiction
- A method of proof that assumes the opposite of what is to be shown, then follows logical steps until a false statement appears. For irrationality of √n, assume √n = p/q in lowest terms and show that p and q must share a factor, contradicting the lowest-terms assumption.
- Composite number
- A positive integer greater than 1 that is not prime. It has at least one positive divisor other than 1 and itself. For example, 15 = 3 × 5 is composite because it can be written as a product of smaller primes.
- HCF–LCM relation
- For two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. This holds only for a pair of numbers, never for three or more together.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| HCF–LCM product identity | HCF(a, b) × LCM(a, b) = a × b | a and b are any two positive integers. Use it to find one value when the other and the product are known. It does not apply to three numbers. |
Practice questions with answers
1. State the Fundamental Theorem of Arithmetic.
Every composite number can be expressed uniquely as a product of primes, apart from the order in which the primes appear. For instance, 156 = 2² × 3 × 13, and no other set of primes will multiply to give 156.
2. Find the HCF and LCM of 510 and 92 by prime factorisation and verify that HCF × LCM equals the product of the two numbers.
Prime factors are 510 = 2 × 3 × 5 × 17 and 92 = 2² × 23. The only common prime is 2, with lowest index 1, so HCF = 2. For LCM, take each prime with its largest index: 2² × 3 × 5 × 17 × 23 = 23460. Then HCF × LCM = 2 × 23460 = 46920, and 510 × 92 = 46920, so the relation holds.
3. Find the HCF and LCM of 144, 198 and 252 by prime factorisation.
Prime factorisations are 144 = 2⁴ × 3², 198 = 2 × 3² × 11 and 252 = 2² × 3² × 7. HCF uses primes common to all with the smallest index: 2¹ × 3² = 18. LCM uses every prime appearing with the greatest index: 2⁴ × 3² × 7 × 11 = 11088. So HCF is 18 and LCM is 11088.
4. Check whether 4ⁿ can end with the digit 0 for any natural number n.
No. Since 4ⁿ = (2²)ⁿ, its only prime factor is 2. A number ending in 0 must be divisible by 10 = 2 × 5, so its prime factorisation must contain both 2 and 5. The factor 5 never appears in 4ⁿ, so 4ⁿ can never end in 0.
5. Prove that √5 is irrational.
Assume √5 is rational, so √5 = p/q where p and q are coprime integers and q ≠ 0. Squaring gives p² = 5q², so 5 divides p²; because 5 is prime, 5 divides p. Write p = 5k; then 25k² = 5q², so q² = 5k² and 5 divides q too. Thus p and q share a factor 5, contradicting that they are coprime. Hence √5 is irrational.
6. The HCF of two numbers is 16 and their product is 3072. Find the LCM.
Using the identity HCF × LCM = product of the two numbers, LCM = 3072 / 16 = 192. Therefore the LCM of the two numbers is 192.
7. Can two numbers have 18 as their HCF and 380 as their LCM? Give a reason.
No. The HCF must divide every common multiple of the numbers, so it must divide the LCM exactly. Here 18 does not divide 380, because 380/18 is not an integer. Therefore no pair of numbers can have HCF 18 and LCM 380.
8. Explain why 7 × 11 × 13 + 13 is a composite number.
The expression can be rewritten by taking 13 common: 13(7 × 11 + 1) = 13 × 78. Since 78 is greater than 1, the number has a divisor other than 1 and itself; in fact 13 × 78 = 2 × 3 × 13², so it is composite.
9. Find the least number which, when divided by 12, 16 and 30, leaves the same remainder 5 in each case.
The number must be 5 more than a common multiple of 12, 16 and 30. Their LCM is 2⁴ × 3 × 5 = 240. Therefore the required least number is 240 + 5 = 245.
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