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LCM and HCF

Greatest Common Divisor (GCD)/ Highest Common Factor (HCF)
The highest common factor of two or more numbers is the greatest common divisor, which divides each of those numbers an exact number of times. The process to find the HCF is
  1. Express the numbers given as a product of prime numbers separately i.e. finds factors of numbers
  2. Take the product of prime numbers common to both numbers
Illustration 1: Find the HCF of 1728 and 14.
Sol: The prime factorization of 1728 is 12 × 12× 12= 33x26.
The prime factorization of 14 is 2 × 7.
The common prime factor is 2. HCF = 2.
Illustration 2: Find the HCF of 27, 18 and 36.
Sol: Firstly find the prime factors of the numbers such as 27 = 3 × 3 × 3, 18 = 3 × 3 × 2 and
36 = 3 × 3 × 2 × 2, then take the common prime numbers, which are 3 & 3.
Now the product of these prime numbers i.e. 3 × 3 = 9 is the HCF of these two numbers.

Understanding LCM or Least Common Multiple
The least common multiple (LCM) of two or more numbers is the smallest of the numbers, which is exactly divisible by each of them, e.g. consider two numbers 18 and 24
The multiples of 18 are: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, ....
The multiples of 24 are: 24, 48, 72, 96, 120, 144, 168, 192, 216,......
The common multiples of both 18 and 24 are 72, 144, 216,....
The least common multiple is 72.
Here again try to break the words in reverse order and understand the concept. Firstly find the multiples of the numbers. Secondly, the common multiples of the numbers and finally the least out of those will be the LCM.
The process to find the LCM is
  1. Express the numbers given as a product of prime numbers separately i.e. finds factors of numbers
  2. Take the product of prime factors of the two numbers after eliminating repetition of the common factors.

Tips To solve the Problems:

  • The product of the two numbers is always equal to the product of their HCF and LCM.
  • In case of HCF, if some remainders are given, then firstly those remainders are subtracted from the numbers given and then their HCF is calculated.
  • In case of LCM, if a single remainder is given, then firstly the LCM is calculated and then that single reminder is added in that.
  • In case of LCM, if for different numbers different remainders are given, then the difference between the number and its respective remainder will be equal. In that case, firstly the LCM is calculated, then that common difference between the number and its respective remainder is subtracted from that.
  • Sometimes in case of HCF questions, the required remainder is given and when the remainder is not given, in those cases you will generally have three numbers given. For answering the question, you need to take the difference of the three pairs of numbers, now the HCF of these differences will become the answer e.g. if you have to find the greatest number, which when divides 83, 93 and 113 and leaves the same remainder. Here you will take the three differences i.e. 93 – 83 = 10 ; 113 – 93 = 20 ; 113 – 83 = 30, after that find the HCF of these differences, which comes out to be 10. Now you can check for yourself- when 10 divides these three numbers, the reminder obtained is 3 in each case and that is what the question was asking for.
  • Whenever the question talks about the greatest or maximum, then in most of these cases it will be a question of HCF. Secondly, whenever the question is related to classification or distribution into groups, then in all the cases it is HCF only.
  • Whenever the question talks about the smallest or minimum, then in most of the cases it will be a question of LCM. Secondly, whenever the word ‘together’ or ‘simultaneous’ is used in the question, then in all the cases it is LCM.
  • Before solving the problems on HCF and LCM in the real exam, you must practice some HCF and LCM worksheets.

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