HCF and LCM

The full form of HCF is the highest common factor, and the full form of LCM is the lowest common multiple. The highest common factor (HCF) of more than two numbers is the largest number that divides all the given numbers without leaving any reminder. The lowest common multiple (LCM) is the lowest common multiple of those numbers. HCF and LCM help to solve our day-to-day problems related to grouping and sharing.

HCF and LCM

HCF, also known as the “greatest common divisor”, is the highest common multiplier present among a group of numbers. For example, for the numbers 10 and 20, the HCF (highest common factor) is 10, but for the numbers 10, 15, and 20, the HCF (highest common factor) is 5. Whereas the LCM (lowest common multiple) of the numbers 10 and 20 is 20. The LCM (lowest common multiple) of 9, 12, and 18 is 36. In the case of LCM, let’s assume there are two numbers, 8 and 16. The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, and so on. The multiples of 16 are 16, 32, 48, 64, 80, and so on. The least common multiple is 16. So, the LCM (least common multiple) of these two numbers, 8 and 16, is 16. The method of determining HCF (highest common factor) and LCM (lowest common multiple): The two most common methods of determining LCM and HCF are the Prime Factorization Method and the Division Method. To calculate HCF and LCM using the prime factorization method, first find the prime factor. To calculate the HCF of the numbers, first determine the prime factors of the numbers. For example, to determine the HCF of 50 and 25 by the prime factorization method, The prime factors of 50 are 5 * 5 * 2. The prime factors of 25 are 5 and 5. So, the common factors of 25 and 50 are 5 x 5. So, the HCF of 25 and 50 is 5 x 5 = 25. LCM by prime factorization: To determine the LCM of the numbers, one must find the prime factors. First one needs to make a list of all common factors and then determine the product of all common and uncommon prime factors of the numbers, but the common factors are included only once. For example, the example to determine the LCM of 15 and 30 The prime factors of 15 are = 5 * 3 * 2. The prime factors of 30 are = 5*3*2*2. So the product of all the common and uncommon factors is = 5 * 3 * 2 * 2 = 30. So, the LCM (lowest common multiple) of 15 and 30 is equal to 30. determining HCF (highest common factor) and LCM (lowest common multiple) by the division method- To find the HCF by division, the first one needs to divide the larger number by the smaller number and then note down the remainder. Next, one should use the reminder as to the divisor, and the divisor used before should be used as the dividend, and the division should be performed again. This should continue till the reminder is not equal to 0. The last divisor will be the HCF. For example, let’s determine the HCF (highest common factor) of 180 and 360. First, 360 is divided by 180 and the remainder is 0. So, the HCF of these two numbers is 180. To determine the LCM in this method, let’s first divide the numbers by the smallest possible prime number, and then the same step is repeated and the product is calculated when there is the least possible reminder remains to determine the LCM. For example, to determine the LCM of 7, 14, and 21, all these numbers are divided by 7 and the remaining reminders are 1, 2, and 3. So the product of all these is 7 * 1 * 2 * 3, which equals 42. So the LCM of these numbers 7, 14, and 21 is 42.

The relationship between LCM (Lowest common multiple) and HCF (Highest common factor)

The relationship between HCF and LCM is that the multiplication product of both HCF and LCM is equal to the product of the numbers. For example, the HCF of two numbers, 10 and 20, is 10 and the LCM of the numbers is 20. The product of the numbers is 10*20 = 200 and the product of HCF and LCM is also 10*20 = 200, thus proven.

Problems regarding HCF and LCM

What is the least number that is divisible by 12, 15, and 20? Answer: So, the LCM of the numbers should be determined. The LCM of the numbers is equal to 60. So, the least number that is exactly divisible by 12, 15, and 20 is 60. The HCF of the two numbers is 13. If these two numbers are in the ratio of 15:11, then find the numbers. Answer: Let’s assume the numbers are 15x and 11x. The LCM is 15*11*x. As a result, The product of LCM and HCF is a product of two numbers. 15x 11x 13 = 15x *11x Or, x = 13 So the numbers are 15*13 and 11*13. So, the numbers are 195 and 143.

Conclusion

LCM (Lowest Common Multiple) and HCF (Highest Common Factor) are two very basic concepts in mathematics but also very useful concepts. Aspirants must keep these concepts handy as they have too many practical applications.