This Real Numbers preview covers prime factorisation, HCF-LCM applications and irrationality. The 2026-27 CBSE syllabus lists the Fundamental Theorem of Arithmetic and proofs that the square roots of 2, 3 and 5 are irrational. Euclid's algorithm and decimal expansions below are additional foundation revision, not separately listed topics in the current Real Numbers syllabus.

Fundamental Theorem of Arithmetic: every composite positive integer has a prime factorisation that is unique apart from the order of its factors. For positive integers a and b, HCF(a,b) times LCM(a,b) equals ab. Use the smallest prime powers for HCF and the largest for LCM.

Irrationality proof template: let p be 2, 3 or 5, each of which is prime. Suppose $\sqrt{p}=a/b$ for positive coprime integers a,b. Then $a^2=p{b}^2$, so p divides a. Write $a=pk$. Substitution gives $b^2=p{k}^2$, so p also divides b. This contradicts coprimality. Therefore $\sqrt{2}$, $\sqrt{3}$ and $\sqrt{5}$ are irrational.

Application: if $3+2\sqrt{5}$ were rational, subtracting 3 and dividing by 2 would make $\sqrt{5}$ rational, contradicting the proof above. Thus $3+2\sqrt{5}$ is irrational.

Decimal expansion revision: first reduce a rational number to a fraction with a positive denominator. Its decimal terminates exactly when that denominator is $2^m{5}^n$ with non-negative integers m,n; a denominator of 1 is included.

Mini Quiz: Attempt these without seeing the solution. Give yourself 8 minutes.

Solutions: Q12: $65=5\times 13$ and $117=9\times 13$, so HCF is 13. Q13: $12=2^2\times 3$ and $18=2\times 3^2$, so LCM is 36. Q14: $40=2^3\times 5$ and the fraction is reduced, so the decimal terminates. Q15: $7$ is not a perfect square, so $\sqrt{7}$ is irrational.