HCF and LCM - Shortcuts & Tricks for 2026 - 2027 Placement Tests, Job Interviews & Exams

HCF and LCM are fundamental concepts in quantitative aptitude, frequently appearing in placement tests, bank exams, and MBA entrance exams. A strong grasp of these concepts is crucial for improving scores across various aptitude topics. This tutorial offers tips and tricks for quickly identifying and solving HCF and LCM problems. **Highest Common Factor (HCF)** HCF, also known as the Greatest Common Divisor (GCD), is the largest number that can divide two or more numbers without leaving a remainder. **Method for Calculating HCF:** To find the HCF of numbers like 42, 54, and 36, you systematically divide them by common prime factors until no further common division is possible. 1. **Divide by 2:** All numbers (42, 54, 36) are divisible by 2, yielding 21, 27, and 18. 2. **Divide by 3:** The new set of numbers (21, 27, 18) are all divisible by 3, yielding 7, 9, and 6. 3. **Stop:** At this point, 7, 9, and 6 do not have a common factor other than 1. Although 9 and 6 are divisible by 3, 7 is not. For HCF, all numbers must be divisible by the chosen factor. 4. **Multiply the common factors:** The HCF is the product of the common factors found on the left side: 2 * 3 = 6. Thus, the HCF of 42, 54, and 36 is 6. This method is straightforward: keep dividing by common factors until you can't divide all numbers further, then multiply the divisors. **Least Common Multiple (LCM)** LCM is the smallest positive integer that is a multiple of two or more numbers. **Method for Calculating LCM:** To find the LCM of numbers like 250, 100, and 125, follow a similar division process to HCF, but continue even if only a pair of numbers (or even a single number) can be divided. 1. **Divide by 5:** All numbers (250, 100, 125) are divisible by 5, yielding 50, 20, and 25. 2. **Divide by 5 again:** All numbers (50, 20, 25) are divisible by 5, yielding 10, 4, and 5. 3. **Divide by 2 (pair):** 10 and 4 are divisible by 2. This yields 5, 2, and 5 (the number 5 that wasn't divisible by 2 is carried down). 4. **Divide by 5 (pair):** The two 5s are divisible by 5. This yields 1, 2, and 1 (the number 2 that wasn't divisible by 5 is carried down). 5. **Divide by 2 (single):** The remaining 2 is divisible by 2. This yields 1, 1, and 1. 6. **Multiply all divisors:** The LCM is the product of all divisors on the left side: 5 * 5 * 2 * 5 * 2 = 500. The key difference from HCF is that you continue dividing until all numbers become 1, carrying down numbers that are not divisible by the current factor. **Tips and Tricks for HCF and LCM:** **Quick HCF by Observation:** For numbers like 18, 24, 30, you might quickly observe that all are divisible by 6 (18=6*3, 24=6*4, 30=6*5). Since 3, 4, and 5 have no common factors, 6 is the HCF. Similarly, for 24, 36, 48, you might recognize them as multiples of 12, making 12 the HCF. Practice helps in recognizing these patterns. **Quick LCM by Elimination:** For numbers like 12, 36, 72, 144: 1. If one number is a multiple of another, eliminate the smaller number. 2. 36 is a multiple of 12, so eliminate 12. 3. 72 is a multiple of 36, so eliminate 36. 4. 144 is a multiple of 72, so eliminate 72. The largest remaining number, 144, is the LCM. Another method for LCM: For 36 and 48, take the largest number (48) and check its multiples: - 48 * 1 = 48 (not divisible by 36) - 48 * 2 = 96 (not divisible by 36) - 48 * 3 = 144 (divisible by 36, so 144 is the LCM). This method is best for smaller or familiar numbers. For larger numbers, the division method is generally more reliable. **LCM for Fractions:** When dealing with fractions, such as 1/30 - 1/45, you can find the LCM of the denominators (30 and 45) or simply multiply them (30 * 45 = 1350) and then adjust the numerators. For instance, 45/1350 - 30/1350 = 15/1350 = 1/90. This direct multiplication of denominators works well, especially when numerators are 1. If numerators are different, like 4/30 - 2/45, the same principle applies: (4 * 45 - 2 * 30) / (30 * 45) = (180 - 60) / 1350 = 120 / 1350. **HCF and LCM of Fractions:** - **LCM of fractions:** LCM of Numerators / HCF of Denominators. - **HCF of fractions:** HCF of Numerators / LCM of Denominators. Example: Find LCM of 36/225, 48/150, 72/65. 1. LCM of numerators (36, 48, 72) = 144. 2. HCF of denominators (225, 150, 65) = 5. (All are divisible by 5, then 45, 30, 13 have no common factors). 3. LCM of fractions = 144/5. **Problem Solving with HCF and LCM:** **Identifying HCF vs. LCM Problems:** - **"Greatest number," "largest number," "maximum size":** Indicates HCF/GCD. - **"Least number," "smallest number," "total number," "when will they meet again":** Indicates LCM. **Question 3: Greatest number dividing 17, 42, 93 and leaving remainders 4, 3, 15.** 1. Since we need the "greatest number," it's an HCF problem. 2. Subtract the remainders from the original numbers to get numbers that are exactly divisible: - 17 - 4 = 13 - 42 - 3 = 39 - 93 - 15 = 78 3. Find the HCF of 13, 39, 78. - All are divisible by 13 (13*1, 13*3, 13*6). - 1, 3, 6 have no common factors. 4. HCF = 13. The greatest number is 13. **Question 4: Least number divided by 36, 24, 16 leaving 11 as remainder.** 1. Since we need the "least number," it's an LCM problem. 2. Find the LCM of 36, 24, 16. - Divide by 4: 9, 6, 4 - Divide by 3 (for 9, 6): 3, 2, 4 (carry 4) - Divide by 2 (for 2, 4): 3, 1, 2 (carry 3) - No common factors for 3, 1, 2. - LCM = 4 * 3 * 2 * 3 * 2 = 144. 3. Since the remainder is the same (11) in each case, add it to the LCM: 144 + 11 = 155. The least number is 155. **Question 5: Least number divided by 20, 48, 36 leaving remainders 13, 41, 29.** 1. It's an LCM problem ("least number"). 2. Notice the difference between each number and its remainder: - 20 - 13 = 7 - 48 - 41 = 7 - 36 - 29 = 7 The difference is common (7). 3. Find the LCM of 20, 48, 36. - Divide by 4: 5, 12, 9 - Divide by 3 (for 12, 9): 5, 4, 3 (carry 5) - No common factors for 5, 4, 3. - LCM = 4 * 3 * 5 * 4 * 3 = 720. 4. Subtract the common difference from the LCM: 720 - 7 = 713. The least number is 713. **Question 6: Least possible four-digit number divided by 12, 16, 18, 20 leaving 21 as remainder.** 1. It's an LCM problem ("least number"). 2. Find the LCM of 12, 16, 18, 20. - Divide by 2: 6, 8, 9, 10 - Divide by 2: 3, 4, 9, 5 - Divide by 3: 1, 4, 3, 5 (carry 4, 5) - No common factors for 1, 4, 3, 5. - LCM = 2 * 2 * 3 * 4 * 3 * 5 = 720. 3. The LCM (720) is a three-digit number, but we need the least *four-digit* number. 4. Find multiples of 720 until you get a four-digit number: - 720 * 1 = 720 (three-digit) - 720 * 2 = 1440 (four-digit, this is the least four-digit multiple). 5. Add the remainder (21) to this multiple: 1440 + 21 = 1461. The least possible four-digit number is 1461. **Question 7: Ratio of two numbers is 5:6, LCM is 480. Find HCF.** 1. Key concept: Product of two numbers (A * B) = Product of their HCF and LCM. 2. Let the numbers be 5k and 6k, where k is the common factor. 3. HCF of 5k and 6k is k (since 5 and 6 are coprime). 4. Apply the formula: (5k) * (6k) = k * 480 5. 30k² = 480k 6. Since k cannot