LCM Formulas

September 9, 2023

Formulas for Lowest Common Multiple (LCM)

This page consists of  LCM Formulas its Definition and Methods to calculate LCM.  LCM stands for Least Common Multiple or Lowest Common Multiple. The Lowest Common Multiple is the smallest positive integer that is evenly divisible by both two or more integers. 

LCM Formula

Formulas for LCM

Formulas to calculate LCM

  • Prime factorization
    • We multiply the prime factors with the highest powers to get the LCM.
    • For example: Find the LCM of 12 and 30
    • Prime factorization of 12 = 2 * 2 * 3 = 22 * 31 * 50
    • Prime factorization of 30 = 2 * 3 * 5 = 21 * 31 * 51
    • Highest exponent value we take 22 * 31 * 51 = 60
    • Therefore, LCM (12, 30) = 60
  • Line division

    • The LCM (12, 30) = 22 × 31 × 51 = 60

Important Points

  • The product of two numbers’ L.C.M. and H.C.F. is the same as the product of the numbers. The L.C.M. of 6 and 12 is 12, and the H.C.F. of 6 and 12 is 6. We see that the product of 6 and 12 is also the product of 6 and 12 L.C.M. and H.C.F.
  • Properties of L.C.M:
    • L.C.M is associative.
    • L.C.M is commutative.
    • L.C.M is distributive.
  • Greatest Common Factor (GCF): It’s important to remember what a factor of a number is. A factor is a number that evenly divides another number.
    • Here, 1,2,3,4,6,8, 12, and 24 are all possible divisions of 24.
    • As a result, 1,2,3,6, 8,12, and 24 are all factors of 24.
      1,2,3,4,6,9,12,18, and 36 are all possible divisions of 36.
    • As a result, factors of 36 are 1,2,3,4,6,9,12, and 36.
      As a result, the greatest common factor of 24 and 36 is 12.
  • The LCM (Lowest Common Multiple) of two or more numbers is the smallest of their common multiples. Several methods exist for determining the LCM of two or more numbers. LCM is defined to be zero if either a or b is zero.

Question and Answer for LCM

Question : In annual examination of G.D. goenka school, a question was asked from maths topic as follows: The least number, which when divided by 12, 15, 20 or 54 leaves a remainder of 4 in each case, is :

A. 450
B. 454
C. 540
D. 544

Solution:

LCM of 15, 12, 20, 54 = 540
Then number = 540 + 4 = 544
[4 being remainder]

Question : Akbar, Birbal, Chanakya start running at the same time and at the same point in the same direction in a circular stadium. Akbar completes a round in 252 seconds, Birbal in 308 seconds and Chanakya in 198 seconds. After what time will they meet again at the starting point ?

A. 26 minutes 18 seconds
B. 42 minutes 36 seconds
C. 45 minutes
D. 46 minutes 12 seconds

Solution :

Required time = LCM of 252, 308 and 198 seconds
LCM = 2 \times 2 \times 7 \times9 \times11
= 2772 seconds
= 46 minutes 12 seconds

Question : Find the greatest number of four digits which, when divided by 15, 19, 21, and 28, leaves remainders of 8, 2, 16, and 12, respectively.

A. 3521
B. 3536
C. 3532
D. 3569

Solution :

To find the required number,
we need to find the greatest four-digit number that is divisible by
the LCM of 15, 19, 21, and 28 is 3,570.
Subtracting the sum of remainders from the result: 3,570 – 38 = 3,532.

Question : Let the least number of six digits which when divided by 4, 6, 10, 15 leaves in each case same remainder 2 be N. The sum of digits in N is :

A. 3
B. 5
C. 4
D. 6

Solution :

LCM of 4, 6, 10, 15 = 60
Least number of 6 digits = 100000
Dividing 100000 by 60 get we get 40
The least number of 6 digits which is exactly divisible by 60
= 100000 + (60 – 40)
= 100020
\therefore Required number (N)
= 100020 + 2 = 100022
Hence, the sum of digits = 1 + 0+ 0 + 0 + 2 + 2 = 5

Question : Find that least number which when doubled will be exactly divisible by 12, 18, 21 and 30 ?

A. 2520
B. 1260
C. 630
D. 196

The LCM of 12, 18, 21, 30
LCM = 2 \times 3 \times 2 \times 3 \times 7 \times 5
= 1260
\therefore The required number = 1260/2 = 630

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