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Math18

Least Common Multiple - lcm

Content

What is lcm?

The multiples that several natural numbers have in common, the least of these multiples is known as the least common multiple, also abbreviated with lowercase letters such as “lcm”. In other words, the least common multiple is interpreted as the smallest number of the common multiples.


How to get the least common multiple?

To obtain the lcm it is advisable to draw a vertical line, to the left of the line are the numbers or the number from which we want to obtain the least common multiple.

The right of the line we will put the prime number that will divide the number from which we want to obtain the lcm.

We are going to make a simple example, it is only necessary to obtain the lcm of a number, calculating the lcm of 18, below is shown step by step to solve the problem:

  1. First, a vertical line is drawn and on the left we will place the number from which we want to obtain the lcm, in this case it is 18, it is very important to have an order.

    18   

          

          

          

    Note: A line was drawn in order to visualize each stage of the operations.

  2. It is checked if the number to be analyzed is divisible by 2, if it is divisible then we put the number 2 to the right of the line and below 18 we put the result of the division.

    18  2

      9  

          

          

  3. Again we must check if the number to be analyzed, now 9, is divisible by 2, since it is not divisible then we proceed to the next prime number that corresponds to 3, since 9 if it is divisible by 3 we can continue with the analysis.

    18  2

      9  3

      3  

          

  4. Again we verify if the number to be analyzed, now 3, is divisible by 3, since if it is divisible then we continue with the analysis.

    18  2

      9  3

      3  3

      1  

  5. When the number to analyze is 1 it means that we can already obtain the lcm considering a multiplication of the numbers that are to the right of the line.
    2 x 3 x 3 = (18) ← mcm

    Note: Since we are only obtaining the lcm of a number, the result of the operation is going to be the same number, therefore, the lcm of 18 is 18.

    Examples:

    A)

    20  2

    10  2

      5  5

      1  

    B)

    7  7

    1  

    C)

    25  5

      5  5

      1  

    D)

    16  2

      8  2

      4  2

      2  2

      1  

    The lcm is:
    A) 2 x 2 x 5 = 20
    B) 7
    C) 5 x 5 = 25
    D) 2 x 2 x 2 x 2 = 16

    Exercise:

    What is the least common multiple? lcm =?
    A)

    9    = ?

    B)

    16    = ?

    C)

    45    = ?

    D)

    82    = ?

We are going to make another example of 2 or more numbers from which it is required to obtain the least common multiple, the procedure is the same as for a number. Considering as an example to obtain the lcm of 5 and 4.

  1. First we must draw a vertical line and put a separation between the numbers to analyze.

    5    4  

              

              

              

    Note: A line was drawn to visualize each stage of the operations.

  2. It is checked if the numbers to be analyzed are divisible by 2, we observe that 5 cannot be divided by two since an integer is not obtained, but the number 4 can be divided by 2, therefore, the number 5 becomes to place in the next row and with the number 4 we make the corresponding division.

    5    4  2

    5    2  

              

              

  3. Again we must check if the number to be analyzed, now 5 and 2, are divisible by 2, again 5 is not divisible by two but to number 2 we can divide it by 2.

    5    4  2

    5    2  2

    5    1  

              

  4. Again we verify if the number to be analyzed, now 5 and 1, are divisible by 2, as one of the numbers to be analyzed has already reached 1 we will only concentrate on the number 5, since 5 is not divisible by 2 we go to the next prime number which is 3, but 5 is not divisible by 3 either, since 5 is a prime number so let's divide by 5.

    5    4  2

    5    2  2

    5    1  5

    1    1  

  5. When the numbers to be analyzed are 1 it is already possible to obtain the lcm considering a multiplication of the numbers to the right of the line.
    2 x 2 x 5 = 20
    The lcm of 5 and 4 is 20.

    Examples:

    A)

    8    6  2

    4    3  2

    2    3  2

    1    3  3

    1    1  

    B)

    15    18  2

    15      9  3

      5      3  3

      5      1  5

      1      1  

    C)

    24    33  2

    12    33  2

      6    33  2

      3    33  3

      1    11  11

      1      1  

    D)

    39    17  3

    13    17  13

      1    17  17

      1      1  

    The lcm is:
    A) 2 x 2 x 2 x 3 = 24
    B) 2 x 3 x 3 x 5 = 90
    C) 2 x 2 x 2 x 3 x 11 = 264
    D) 3 x 13 x 17= 663

    Exercise:

    What is the least common multiple? lcm =?
    A)

      6    13    = ?

    B)

      9    22    = ?

    C)

      24    17    = ?

    D)

      45    25    = ?


Calculation of the lcm in a fraction

Assuming we have the following sum of fractions:

1 / 4
+
1
/ 12

We can get the denominator by multiplying 4 x 12 = 48, but it would be a very large number and the fraction would not be simplified, that is why we are going to get the lcm of 4 and 12.

Following the procedure to obtain the lcm the result was the following:

4   12  2

2     6  2

1     3  3

1     1  

The lcm of 4 and 12 is:

2 x 2 x 3 = 12

Now that we know the lcm, the operation to be performed with fractions is simplified:

1 / 4
+
1
/ 12
=
3 + 1
/
12
=
4
/
12

The resulting fraction 4/12 can be simplified by dividing the numerator and denominator by 4, therefore, 4/12 = 1/3. This can be verified by performing both divisions 4/12 = 0.3333 and 1/3 = 0.3333.