HCF and LCM Aptitude Tricks HCF ShortcutShort Tricks HCF Kaise Nikale Part 2

Mastering HCF and LCM: Smart Shortcuts and Techniques

HCF (Highest Common Factor) and LCM (Least Common Multiple) are fundamental concepts in number theory that every CBSE Class 10 student must master. These concepts are introduced early in mathematics education but become critically important in the chapter on Real Numbers, where the Fundamental Theorem of Arithmetic and Euclid's Division Lemma rely heavily on your ability to find factors and multiples efficiently. Whether you are solving textbook exercises or appearing in competitive examinations, knowing both the standard methods and clever shortcuts can save valuable time.

The most reliable method for finding HCF is the prime factorisation method, where you break down each number into its prime factors and identify the common ones. For LCM, you take the highest power of each prime factor present across all numbers. However, there are several faster techniques. The division method (also called the long division method) is particularly useful when dealing with large numbers, as it avoids writing out lengthy factor trees. Another powerful approach is using the relationship: HCF × LCM = Product of Two Numbers. This formula works only for two numbers and allows you to find one value instantly if you know the other three. For three or more numbers, you must apply methods step by step in pairs.

Competitive exam shortcuts include the difference method, where if the difference between two numbers is small, you can check only the factors of that difference to find the HCF. For example, the HCF of 84 and 72 must be a factor of 12 (their difference). Another useful trick involves working with remainders: if two numbers leave the same remainder when divided by a common divisor, that divisor is a factor of their difference. Understanding these patterns helps you eliminate options quickly in multiple-choice questions without performing full calculations. Continuous division using Euclid's algorithm — repeatedly dividing the larger number by the smaller and replacing until the remainder becomes zero — is both systematic and fast for competitive problem-solving.

• HCF is the greatest number that divides all given numbers exactly, while LCM is the smallest number that is divisible by all given numbers.
• The formula HCF × LCM = Product of Two Numbers holds true only for two numbers, not for three or more.
• The division method is faster than prime factorisation when dealing with large numbers or numbers with many digits.
• If the difference between two numbers is small, the HCF must be a factor of that difference — use this to check options quickly.
• For finding HCF of fractions, divide the HCF of numerators by the LCM of denominators; for LCM of fractions, divide the LCM of numerators by the HCF of denominators.
• Euclid's Division Lemma states that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b — this forms the basis of the division method.

A strong command over HCF and LCM problem-solving directly supports your performance in the Real Numbers chapter and also prepares you for aptitude-based questions in future competitive exams like CUET, SSC, and banking entrance tests. Practice applying these shortcuts with a variety of problems from your NCERT textbook and previous years' question papers to build both speed and accuracy.