Introduction
Almost every competitive exam in Bangladesh — BCS (preliminary and written), Bank recruitment (Bangladesh Bank AD, Combined bank Officer/SO), and IT-officer/computer posts — opens its mathematics section with the Number System and its two most exam-heavy tools: HCF (GCD) and LCM. These topics look elementary, but examiners love them because a single well-chosen question can test factorisation, divisibility, remainders, ratios and even word-problem reasoning at once. If you master this chapter you will comfortably answer 3–6 marks in any typical exam, and you will move faster through later chapters (fractions, ratio, time and work, pipes and cisterns) that silently depend on LCM.
This lesson builds everything from first principles: what a number is, how numbers are classified, the rules that let you test divisibility in seconds, and then a complete, method-by-method treatment of HCF and LCM with the exact shortcuts examiners expect.
Learning objectives
By the end of this lesson you should be able to:
- Classify any given number (natural, whole, integer, rational, irrational, real, prime, composite, even, odd).
- Apply divisibility rules for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 instantly.
- Find the prime factorisation of any number and use it as the foundation for HCF and LCM.
- Compute HCF by prime factorisation and by the division (Euclidean) algorithm.
- Compute LCM by prime factorisation and by the common-division method.
- Use the identity HCF × LCM = product of the two numbers, and know when it applies.
- Solve classic exam word problems: greatest measuring length, bells tolling together, remainder problems, and fitting-tiles problems.
- Avoid the standard traps examiners set around remainders and co-prime numbers.