Class 10 · Chapter 1

Real Numbers

Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, and the irrationality of numbers like $\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$ .

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Detailed theory

Given positive integers

a

and

b

, there exist unique whole numbers

q

and

r

such that

a = bq + r,\; 0 \le r < b

. This lemma is the foundation of the Euclidean algorithm for finding HCF. Apply the lemma repeatedly to

(a,b) \to (b,r) \to \ldots

until

r = 0

; the last non-zero remainder is

\text{HCF}(a,b)

Key concepts & formulas

Euclid's division lemma:

a = bq + r,\; 0\le r<b

\text{HCF}\times\text{LCM} = ab

for two positive integers.

Every composite number has a unique prime factorisation (up to order).

\sqrt{2}, \sqrt{3}, \sqrt{5}

are irrational.

\dfrac{p}{q}

has terminating decimal iff

q = 2^a 5^b

(in lowest terms).

p

is prime and

p \mid a^2

, then

p \mid a