Class 10 › Mathematics › Ch 1
Real Numbers
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In 30 seconds
- Every composite number breaks into prime factors in exactly one way — that's the Fundamental Theorem of Arithmetic.
- HCF = product of the smallest powers of common primes; LCM = product of the greatest powers of all primes involved.
- For any two positive integers, HCF × LCM = product of the numbers — for two numbers only, not three.
- √2, √3, √5 are irrational, proved by contradiction using the theorem: if a prime p divides a², it divides a.
- A rational combined with an irrational (added, subtracted, or multiplied by a non-zero rational) stays irrational.
The Fundamental Theorem of Arithmetic
Primes are the atoms of the whole numbers. The Fundamental Theorem of Arithmetic (FTA) says that every composite number is built from those atoms in exactly one way.
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. Example: 3825 = 3² × 5² × 17 — and no other combination of primes will ever multiply to 3825.
To factorise, keep dividing by the smallest prime that fits: 156 = 2 × 78 = 2 × 2 × 39 = 2² × 3 × 13.
A classic application: can a number of the form 4ⁿ end with the digit 0? A number ending in 0 must have both 2 and 5 as prime factors. But 4ⁿ = 2²ⁿ has only 2s — by uniqueness of prime factorisation, 5 can never appear. So no power of 4 ends in zero.
HCF and LCM by prime factorisation
Once two numbers are written as products of primes, their HCF and LCM can be read straight off the factorisations.
- HCF = product of the smallest power of each common prime factor.
- LCM = product of the greatest power of every prime factor that appears in either number.
- The HCF always divides the LCM.
When to use: two of the four quantities are known and you need a third — valid for two numbers only
Find the HCF and LCM of 96 and 404 by prime factorisation, and verify the relation.
Factorise: 96 = 2⁵ × 3 and 404 = 2² × 101.
The only common prime is 2, and its smallest power is 2², so HCF = 4.
LCM takes every prime at its highest power: 2⁵ × 3 × 101 = 9696.
Check: HCF × LCM = 4 × 9696 = 38784, and 96 × 404 = 38784. ✓
Answer: HCF = 4, LCM = 9696; and 4 × 9696 = 38784 = 96 × 404 ✓
Word problems use the same two ideas in disguise: “the largest ribbon that measures both lengths exactly” is an HCF; “the first time the bells ring together again” is an LCM.
Proving irrationality
A rational number can be written as p/q with p, q integers and q ≠ 0. An irrational number cannot. The exam-standard proofs all rest on one theorem and one technique.
Theorem (prime divides square ⇒ prime divides number)
If p is a prime and p divides a², then p divides a itself. This is the engine of every irrationality proof in this chapter, and it follows from the uniqueness part of the FTA.
Proof that √2 is irrational (by contradiction). Suppose √2 were rational: √2 = a/b with a, b coprime integers (the fraction fully reduced). Squaring, 2b² = a², so 2 divides a², and by the theorem 2 divides a. Write a = 2c. Then 2b² = 4c², so b² = 2c², meaning 2 also divides b. But now 2 divides both a and b — contradicting that they were coprime. So √2 cannot be rational. ∎
The same argument works word-for-word for √3 and √5.
Prove that 3 + 2√5 is irrational.
Suppose 3 + 2√5 were rational. Rationals are closed under subtraction and division by a non-zero rational, so √5 = ((3 + 2√5) − 3) / 2 would also be rational. But √5 is irrational — contradiction. Hence 3 + 2√5 is irrational. ∎
Answer: Assuming it rational forces √5 to be rational — contradiction; hence 3 + 2√5 is irrational.
- Applying HCF × LCM = product to three numbers — the relation holds for two numbers only.
- Forgetting to state that a and b are coprime when starting the √2 proof — the contradiction depends on it.
- Treating 1 as a prime (it isn’t — otherwise unique factorisation would fail).
- In “prove 3 + 2√5 is irrational”, skipping the closure argument and just asserting the result — the two-line reduction to √5 is the actual proof.
- Missing a prime factor when computing the LCM — every prime from either number must appear, at its highest power.
Quick revision cards
Fundamental Theorem of Arithmetic?
HCF from prime factorisations?
LCM from prime factorisations?
HCF–LCM relation?
Key theorem behind irrationality proofs?
Why can't 4ⁿ end in 0?
Shape of the √2 proof?