Find the prime factorisation of 72.

2³ × 3²

72 = 2 × 36 = 2 × 2 × 18 = 2 × 2 × 2 × 9 = 2³ × 3².

Understanding Number Theory questions

Number theory questions are about the structure of whole numbers — factors, multiples, primes and divisibility. They are short to state and quick to solve once you know the standard tests.

How to approach it

1. Break the numbers into prime factors.
2. Use the factorisation to read off HCF (common primes, lowest powers) or LCM (all primes, highest powers).
3. Apply divisibility tests instead of long division where possible.
4. Sanity-check that HCF × LCM equals the product of the two numbers.

Common mistakes to avoid

• Swapping the rules for HCF and LCM.
• Treating 1 as a prime number.
• Stopping the prime factorisation too early.

Where it appears: Frequent in CBSE Class 6–10 and in the quantitative sections of competitive exams.