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 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.

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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.

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 ?

For Answers, see at the bottom of the page.

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 = k_{1}n and y = k_{2}n, where k_{1} and k_{2} do not have any common factors.

Then, their L.C.M. = k_{1}k_{2}n

Product of L.C.M. and G.C.F. = (k_{1}k_{2}n)n = k_{1}k_{2}n^{2}....(i)

Product of the two numbers = xy = (k_{1}n)(k_{2}n) = k_{1}k_{2}n^{2}....(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.

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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