LEAST COMMON MULTIPLE - COMPUTING IT MADE EASY, WORD PROBLEMS, EXAMPLES, EXERCISES

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Please study  Division before least common multiple,
if you have not already done so.

It is a prerequisite here.

When two or more numbers are multiplied, then
the product is a multiple of each one of the numbers.

For Example, 2 x 3 x 5 = 30.
30 is a multiple of 2, 3 and 5.

Consider the mutiples of two numbers.
Multiples of 4 = 1 x 4, 2 x 4, 3 x 4, 4 x 4, 5 x 4, 6 x 4, ......
= 4, 8, 12, 16, 20, 24, .....
Multiples of 6 = 1 x 6, 2 x 6, 3 x 6, 4 x 6, 5 x 6, 6 x 6, ......
= 6, 12, 18, 24, 30, 36, .....

From the above, common multiples of 4 and 6 = 12, 24, ....

Of these, 12 is the least.

In Arithmetic and Elementary Number Theory,
it is significant to find the smallest natural number
that is a multiple of two or more numbers.

Since it is a multiple, it can be divided by the two
or more numbers without a remainder.

For example,
when adding or subtracting vulgar fractions (common fractions),
it is useful to find the smallest common multiple of the denominators,
often called the lowest common denominator.

To see an example, to subtract 1/6 from 2/9, we need to know
that 18 is the lowest common multiple of 6 and 9.

Then, we can find 2⁄9 - 1⁄6 = 4⁄18 - 3⁄18 = (4 - 3)⁄18 = 1⁄18.

How do we find that 18 is the smallest common multiple of 6 and 9 ?

To list out the multiples of given numbers and
to find the least of them, as given above
is a tedious process and is very difficult
when the numbers are large.

One better way is by determining the prime factorizations
of the two numbers and comparing factors.

Another simple method is common division method.

The smallest positive integer
that is a multiple of two or more numbers is called
the LEAST COMMON MULTIPLE (L.C.M.)
of the two or more numbers.

The first method of finding L.C.M. is discussed in
Prime Factorization.

The second method called Common Division Method
is discussed here.

Let us again define L.C.M.

Least Common Multiple, L.C.M.

Least Common Multiple, L.C.M. of two or more numbers
is the smallest natural number which is
a multiple of each of the given numbers.

We may note that,

Of two numbers, if one is a multiple of the other,
the larger number is the L.C.M. of the two numbers.

If two numbers are  co-primes,  their L.C.M. is their product.





Common Division Method for finding L.C.M.of two or more numbers

STEP 1 :

Arrange the given numbers in a row in any order.

STEP 2 :

Divide by a number which divides exactly at least two of the given
numbers and co-prime to the other numbers
and carry forward the numbers which are not divisible.

STEP 3 :

Repeat the above process till no two of the numbers are divisibleby same number other than 1.

STEP 4 :

The product of the divisors and the undivided numbers is the requiredL.C.M. of the given numbers.

All these steps are shown at one place as a single unit similar to
Synthetic Division.
The method will be clear by the following examples.

Solved Example 1 of Least Common Multiple :

Find the L.C.M. of the numbers 24, 36 and 40 .

Solution :

Look at the presentation given below.

Step 1 : We arrange the given numbers in a row.

Step2 : 4 is chosen to as divisor.
( not 6 or 12 why ? Though 6 or 12 divide 24 and 36,
they are not co-prime to 40. They share a common factor 2 with 40.)

Division with 4 gave 6, 9, 10 as quotients
and there is no undivided number.

Then 3 is chosen as divisor as it divides 6 and 9 and co-prime to 10.

Division with 3 gave 2, 3 and 10 is undivided number.

Then 2 is chosen as divisor as it divides 2 and 10 and co-prime to 3.

Division with 2 gave 1, 3 is undivided number, and 5.

Step 3 : The process is stopped as no two of 1, 3, 5
are divisible by same number other than 1.

Step 4: L.C.M. = The product of the divisors and the undivided numbers
= 4 x 3 x 2 x 3 x 5 = 360.

The process is shown below.


        4|    24     36     40
        -----------------------
        3|     6      9     10
        -----------------------
        2|     2      3     10
        -----------------------
               1      3      5

L.C.M. = 4 x 3 x 2 x 3 x 5 = 360. Ans.

Solved Example 2 of Least Common Multiple :

Find the L.C.M. of the numbers 15, 24, 30 and 40.

Solution :


        5|    15     24     30     40
       -------------------------------
        3|     3     24      6      8
       -------------------------------
        2|     1      8      2      8
       -------------------------------
        4|     1      4      1      4
       -------------------------------
               1      1      1      1

L.C.M. = 5 x 3 x 2 x 4 = 120. Ans.

Solved Example 3 of Least Common Multiple :

Find the L.C.M. of the numbers 22, 54, 108, 135, 198.

Solution :


        9|    22     54     108     135     198
       -----------------------------------------
       11|    22      6      12      15      22
       -----------------------------------------
        2|     2      6      12      15       2
       -----------------------------------------
        3|     1      3       6      15       1
       -----------------------------------------
               1      1       2       5       1

L.C.M. = 9 x 11 x 2 x 3 x 2 x 5 = 5940. Ans.













Solved Example 4 of Least Common Multiple :

Find the L.C.M. of the numbers 288, 432, 486.

Solution :


        9|    288     432     486
        --------------------------
        3|     32      48     54
        --------------------------
        2|     32      16     18
        --------------------------
         8|    16      8     9
       --------------------------
                    2     1     9

L.C.M. = 9 x 3 x 2 x 8 x 2 x 9 = 7776. Ans.

Solved Example 5 of Least Common Multiple :

Find the L.C.M. of the numbers 9, 12, 15, 18, 24, and 56.

Solution :


        3|    9     12     15     18    24     56
       ------------------------------------------------
        3|    3      4      5      6     8     56
       ------------------------------------------------
        2|    1      4      5      2     8     56
       ------------------------------------------------
        2|    1      2      5      1     4     28
       ------------------------------------------------
        2|    1      1      5      1     2     14
      ------------------------------------------------
              1      1      5      1     1      7

L.C.M. = 3 x 3 x 2 x 2 x 2 x 5 x 7 = 2520. Ans.

Exercise 1 of Least Common Multiple

Find the L.C.M. of the numbers

(1) 12, 15, 18 and 27.

(2) 15, 20, 35, 45 and 50.

(3) 72, 192 and 240.

(4) 48, 64, 72, 96 and 108.

(5) 28, 36, 42, 54 and 60.

For Answers, see at the bottom of the page.

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Solved Example 6 of Least Common Multiple : Word Problems

What is the least number of children who may be
arranged in rows of 12, 16 or 18 in each row ?

Solution :
The required number is nothing but
the least number exactly divisible by 12, 16 and 18.

That means, The required number = L.C.M of 12, 16 and 18.


        2|   12     16     18
        -----------------------
        3|    6      8      9
        -----------------------
        2|    2      8      3
        -----------------------
              1      4      3

L.C.M. = 2 x 3 x 2 x 4 x 3 = 144

Thus, The required number = L.C.M of 12, 16 and 18. = 144. Ans.

Solved Example 7 of Least Common Multiple : Word Problems

Find the greatest number of five digits which when divided
by 12, 15, 18 and 27 leaves a remainder 3 in each case.

Solution :
Let us first find the least number which
gets exactly divided by 12, 15, 18 and 27.
That means L.C.M of 12, 15, 18 and 27.


        3|    12     15     18     27
       -------------------------------
        3|     4      5      6      9
       -------------------------------
        2|     4      5      2      3
       -------------------------------
               2      5      1      3

L.C.M. = 3 x 3 x 2 x 2 x 5 x 3 = 540

Multiples of 540 give remainder 0 when divided by 12, 15, 18 and 27.

Let us find the multiple of 540 which
is slightly less than a six digit number (100,000)

540 x 185 = 99900

99900 is the greatest number of five digits which when divided
by 12, 15, 18 and 27 leaves a remainder 0 in each case.

To get remainder 3, add 3 to the above number.

Thus, the required number = 99900 + 3 = 99903. Ans.

Solved Example 8 of Least Common Multiple : Word Problems

Three persons, A, B and C run on a round track.
A takes 100 seconds, B takes 110 seconds
and C takes 120 seconds to run a round.
If they start together when do they meet again ?

Solution :
The time period of their meeting again
is nothing but the L.C.M. of 100, 110 and 120.


        10|    100     110     120
        --------------------------
         2|     10      11      12
        --------------------------
                 5      11       6
        

L.C.M. = 10 x 2 x 5 x 11 x 6 = 6600

They meet again after 6600 seconds = 110 minutes = 1 hour 50 minutes.Ans.













Exercise 2 of Least Common Multiple : Word Problems

(1) If the fruits are arranged in groups of 3, 4, 6 or 8,
no fruit is left behind. Find the number of fruits.

(2) Find the greatest number of four digits which is exactly
divisible by each one of the numbers 12, 21, 18 and 28.

(3) Five bells begin to toll together and toll respectively
at intervals of 6, 7, 8, 9 and 12 seconds. After how much
time will they toll together again ?

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The Relationship between G.C.F. and L.C.M.

Product of two numbers = Product of their L.C.M. and G.C.F.

Proof :

Let x and y be the two numbers and n be their G.C.F.

Then, x = k1n and y = k2n,
where k1 and k2 do not have any common factors.

Then, their L.C.M. = k1k2n

Product of L.C.M. and G.C.F. = (k1k2n)n = k1k2n2....(i)

Product of the two numbers = xy = (k1n)(k2n) = k1k2n2....(ii)

From (i) and (ii),
Product of two numbers = Product of their L.C.M. and G.C.F. (proved.)

Solved Example 9 of Least Common Multiple : Relationship between G.C.F. and L.C.M.

Product of two numbers is 6912 and their G.C.F. is 24. Find their L.C.M.

Solution :
L.C.M. = (Product of two numbers)⁄(G.C.F.)
= 6912⁄24 = 288. Ans.

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Solved Example 10 of Least Common Multiple : Relationship between G.C.F. and L.C.M.

The G.C.F. of 324 and 360 is 36. Find their L.C.M. Verify the
result by finding the L.C.M. using common division method.

Solution :
L.C.M. = (Product of two numbers)⁄(G.C.F.)
= (324 x 360)⁄36 = 3240. Ans.

Verification :


        9|    324     360     
        ------------------
        4|     36      40     
        ------------------
                9      10     
        

L.C.M. = 9 x 4 x 9 x 10 = 3240 (verified.)











Exercise 3 of Least Common Multiple : Relationship between G.C.F. and L.C.M.

(1), (2) and (3)
Find the L.C.M. of the numbers in Solved Examples 1, 2 and 3 of
Greatest Common Factor,
using the numbers and the G.C.F. found there.

Also verify the results by finding the L.C.M.
using common division method.

For Answers, see at the bottom of the page.

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Answers to Exercise 1 of Least Common Multiple

(1) 540 (2) 6300 (3) 2880 (4) 1728 (5) 3780

Answers to Exercise 2 of Least Common Multiple : Word Problems

(1) 24 (2) 9828 (3) 8 min 24 sec.

Answers to Exercise 3 of Least Common Multiple : Relationship between G.C.F. and L.C.M.

(1) 240 (2) 360 (3) 3787498