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Calculating Lowest Common Multiple (LCM)

Method 1:
 
Write the numbers as product of prime factors.
 
Find the product of the highest powers of the prime factors, which will be the L.C.M.
 
Note: Don’t repeat any factor while writing the product in step2.
 
Example: Find the L.C.M. of 36, 56, 105 and 108.
 
To find the LCM, first we have to find prime factors —
  1. 36 = 22 × 32; 56 = 23 × 7; 105 = 3 × 5 × 7; 108 = 22 × 33
  2. The L.C.M. must contain every prime factor of each of the numbers. Also it must include the highest power of each prime factor which appears in any of them. So, it must contain 2 or it would not be a multiple of 56, it must contain 3 or it would not be a multiple of 108, it must contain 5 or it would not be a multiple of 105, and it must contain 7 or it would not be a multiple of 56 or of 105. Therefore, the L.C.M. = 23 × 33 × 5 × 7 = 7560

Method 2:
 
This is a quicker method to find the prime factors and hence L.C.M. In this method there can be more than one arrangement for the same of numbers. The steps are:
 
Write the numbers in a row and strike out those numbers which are factors of any other number in the set.

Write the factor on the left hand side which can divide maximum of the numbers.
 
Write in the next row the quotients obtained and also those numbers (as they are) which are not divisible by that factor. You can strike out from any row 1, if it appears.
 
Repeat steps 2 and 3 until we get a set where no two numbers have a common factor or divisor, i.e., all the numbers in the row are prime to each other, though individually they may not be prime numbers.

Multiply all the factors or divisors and the numbers left in the last row. The product gives the L.C.M. of the given numbers.
 
L.C.M. of Decimals
 
To find out the L.C.M. of the given numbers in which decimals are given, first of all we find out the L.C.M. of numbers without decimal. And then we see the numbers in which the decimal is given in the minimum digits from right to left. We put the decimal in our result which is equal to that number of digit.
 
Example
Find out the L.C.M. of 0.16, 5.4, and 0.0098.
Solution
First of all we find out the L.C.M. of 16, 54, 98. Here L.C.M. of 16, 54, 98 is 21168. In numbers 0.16, 5.4, 0.0098, the minimum digits from right to left is 5.4. Here 5.4 the decimals is given of one digit from right to left. So, we put decimal in our result such that: 21168 = 1226.8
 
Example
Find the L.C.M. of 48, 10.8 and 0.140.
Solution
L.C.M. of 48, 108 and 140 is 15120. So, L.C.M. of 48, 10.8 and 0.140 = 1.5120
 
L.C.M. of Fractions
 
If a/b, c/d, e/f be the proper fractions then their L.C.M. is given by

Calculating Highest Common Factor (HCF)

Method 1 (Factorization Method)
 
Factor method has discussed above or, we express each given number as the product of primes. Now we take the product of common factors which is our required H.C.F.
 
Example: Find the H.C.F of 144, 336 and 2016.
 
Factors of 144 = 24 × 32; Factors of 336 = 24 × 3 × 7; Factors of 2016 = 25 × 7 × 32;
 
So, H.C.F. of given numbers = 24 × 3 = 48
 
Method 2 (Division Method)
 
We divide the greater number by the smaller number and find out the remainder.
 
Then divide the first divisor by remainder and find the second remainder.
 
Then divide the second divisor by the second remainder.
 
We repeat this process till no remainder is left. The divisor is our required H.C.F.
 
H.C.F. of Fractions
 
If a/b, c/d, e/f be the proper fractions then their H.C.F. is given by

Important Tips

  • Product of two numbers = L.C.M. × H.C.F.
  • Product of n numbers = L.C.M of n numbers × Product of the HCF of each possible pair
  • If ratio of numbers is a : b and H is the HCF of the numbers Then LCM of the numbers = H x a x b = HCF x Product of the ratios.
  • H.C.F of fractions = H.C.F. of Numerators L.C.M. of Denominators
  • L.C.M. of fractions = L.C.M. of Numerators H.C.F. of Denominators
     
    e.g. For two fractions a/b and c/d.
     
    HCF = H.C.F. of a and c
     
    L.C.M. of b and d, LCM = L.C.M. of a and c
     
    H.C.F. of b and d
  • If HCF (a, b) = H1 and HCF (c, d) = H2, then HCF (a, b, c, d) = HCF (H1, H2).
  • LCM is always a multiple of HCF of the numbers . i.e. LCM = (a number) x HCF
  • To find the GREATEST NUMBER that will exactly divide x, y, z.
     
    Just Find H.C.F. of x, y, and z (greatest divisor).
  • To find the GREATEST NUMBER that will divide x, y and z leaving remainders a, b and c respectively.
     
    Find H.C.F. of (x – a), (y – b) and (z – c).
  • To find the LEAST NUMBER which is exactly divisible by x, y and z.
     
    Find the L.C. M. of x, y and z
  • To find the LEAST NUMBER -which when divided by x, y and z leaves the remainders a, b and c respectively.
     
    Required Number = (L.C.M. of x, y and z) – K (Where K= (x - a) = (y - b) = (z - c)
  • Find the LEAST NUMBER, which when divided by x, y and z leaves the same remainder ‘r’ each case. (L.C.M. of x, y and z) + r.
  • Find the GREATEST NUMBER that will divide x, y and z leaving the same remainder in each case.
     
    H.C.F of (x – y), (y – z) and (z – x).
  • or TWO NUMBERS: FIRST NUMBER * SECOND NUMBER = L.C.M * H.C.F
    1. FIRST NUMBER = 
    2. SECOND NUMBER = 
    3. LCM =  
    4. HCF = 
Rule of finding Dividend
 
Dividend = Divisor* Quotient + Remainder




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