Real numbers include all rational and irrational numbers together. Every number you encounter on the number line — integers, fractions, decimals, and non-terminating non-repeating decimals — is a real number. In Class 10, we study two powerful tools: the Fundamental Theorem of Arithmetic and Euclid's Division Lemma, which let us analyze integers deeply.
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Key Concepts
Euclid's Division Lemma: For any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. This is the basis of the Euclidean algorithm for finding the HCF.
Euclid's Division Algorithm (HCF): To find HCF(a, b), apply the lemma repeatedly until the remainder is 0. The last non-zero remainder is the HCF.
Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes in exactly one way (order disregarded). Example: 360 = 23 x 32 x 5.
- HCF and LCM via Prime Factorisation:
- HCF = product of the lowest powers of common prime factors
- LCM = product of the highest powers of all prime factors
- Relation: HCF(a, b) x LCM(a, b) = a x b
Irrational Numbers: Numbers that cannot be written as p/q (q not 0, p and q integers). Examples: √(2), √(3), pi. Their decimal expansions are non-terminating and non-repeating.
Rational Numbers and Decimal Expansions: A rational number p/q (in lowest terms) has a terminating decimal if and only if q has no prime factors other than 2 and 5.
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Worked Examples
Find HCF(135, 225) using Euclid's division algorithm.
Step 1: 225 = 135 x 1 + 90
Step 2: 135 = 90 x 1 + 45
Step 3: 90 = 45 x 2 + 0
Since remainder is 0, HCF(135, 225) = 45.
Find HCF and LCM of 12 and 18 using prime factorisation.
12 = 22 x 3; 18 = 2 x 32
HCF = 21 x 31 = 6; LCM = 22 x 32 = 36
Check: 6 x 36 = 216 = 12 x 18. Correct!
Prove that √(2) is irrational.
Assume √(2) = p/q in lowest terms (HCF(p,q) = 1).
Then 2 = p2/q2, so p2 = 2q2. Thus p2 is even, so p is even.
Write p = 2k. Then 4k2 = 2q2, so q2 = 2k2, making q even.
Both p and q are even — contradicts HCF(p,q) = 1. Hence √(2) is irrational.
Without actual division, determine if 17/125 is terminating.
125 = 53. The denominator has only the prime factor 5. So 17/125 is a terminating decimal (= 0.136).
Find HCF(6, 20, 36) using prime factorisation.
6 = 2 x 3; 20 = 22 x 5; 36 = 22 x 32
HCF = 21 = 2 (only common prime factor).
The LCM of two numbers is 2520 and their HCF is 6. One number is 54. Find the other.
Other number = (HCF x LCM) / first number = (6 x 2520) / 54 = 15120 / 54 = 280.
Express 3825 as a product of prime factors.
3825 / 3 = 1275; 1275 / 3 = 425; 425 / 5 = 85; 85 / 5 = 17.
3825 = 32 x 52 x 17.
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Key Formulas
Key formulas
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Common mistakes
- Forgetting that HCF x LCM = a x b applies only to two numbers, not three or more.
- Assuming √(p) is irrational for all p — it is rational when p is a perfect square.
- In Euclid's algorithm, always ensure r satisfies 0 ≤ r < b.
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Summary
Real Numbers ties together divisibility, prime factorisation, and irrationality proofs. Master Euclid's algorithm and the Fundamental Theorem and you have a solid foundation for all of Class 10 number theory.