Least Common Multiple
The least common multiple (L.C.M.) of two or more numbers is the smallest number which can be exactly divided by each of the given number.
1. Let us find the L.C.M. of 2, 3 and 4.
Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, ...... etc.
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...... etc.
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, ...... etc.
Common multiples of 2, 3 and 4 are 12, 24, 36, ...... etc.
Therefore, the smallest common multiple or least common multiples of 2, 3 and 4 is 12.
We know that the lowest common multiple or LCM of two or more numbers is the smallest of all common multiples.
2. Let us consider the numbers 28 and 12
Multiples of 28 are 28, 56, 84, 112, …….
Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, …….
The lowest common multiple (LCM) of 28 and 12 is 84.
3. Let us consider the first six multiples of 4 and 6.
The first six multiples of 4 are 4, 8, 12, 16, 20, 24
The first six multiples of 6 are 6, 12, 18, 24, 30, 36
The numbers 12 and 24 are the first two common multiples of 4 and 6. In the above example the least common multiple of 4 and 6 is 12.
Hence, the least common multiple or LCM is the smallest common multiple of the given numbers.
Consider the following.
(i) 12 is the least common multiple (L.C.M) of 3 and 4.
(ii) 6 is the least common multiple (L.C.M) of 2, 3 and 6.
(iii) 10 is the least common multiple (L.C.M) of 2 and 5.
We can also find the L.C.M. of given numbers by their complete factorisation.
4. To find for instance, L.C.M. of 24, 36 and 40, we first factorise them completely.
24 = 2 × 2 × 2 × 3 = 2\(^{3}\) × 3\(^{1}\)
36 = 2 × 2 × 3 × 3 = 2\(^{2}\) × 3\(^{2}\)
40 = 2 × 2 × 2 × 5 = 2\(^{3}\) × 5\(^{1}\)
L.C.M. is the product of highest power of primes present in the factors.
Therefore, L.C.M. of 24, 36 and 40 = 2\(^{3}\) × 3\(^{2}\) × 5\(^{1}\) = 8 × 9 × 5 = 360
The least or lowest common multiple (in short LCM) of two or more numbers is smallest number which is divisible by each of the given numbers.
5. Find the first two common multiples of: 4 and 10
Solution:
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ......
Multiples of 10 are 10, 20, 30, 40......
Thus, the first two common multiples of 4 and 10 are 20 and 40.
6. Find the LC.M. of 12 and 18.
Solution:
Multiples of 12 are 12, 24, 36, 48, 60, 72, 84 ........
Multiples of 18 are 18, 36, 54, 72, 90 .........
Thus, common multiples of 12 and 18 are 36, 72,.....
There, L.C.M. of 12 and 18 is 36.
Solved examples to find the lowest common multiple or the least common multiple:
1. Find the L.C.M. of 8, 12, 16, 24 and 36
8 = 2 × 2 × 2 = 2\(^{3}\)
12 = 2 × 2 × 3 = 2\(^{2}\) × 3\(^{1}\)
16 = 2 × 2 × 2 × 2 = 2\(^{4}\)
24 = 2 × 2 × 2 × 3 = 2\(^{3}\) × 3\(^{1}\)
36 = 2 × 2 × 3 × 3 = 2\(^{2}\) × 3\(^{2}\)
Therefore, L.C.M. of 8, 12, 16, 24 and 36 = 2\(^{4}\) × 3\(^{2}\) = 144.
2. Find the LCM of 3, 4 and 6 by listing the multiples.
Solution:
The multiple of 3 are 3, 6, 12, 15, 18, 21, 24
The multiple of 4 are 4, 8, 12, 16, 20, 24, 28
The multiple of 6 are 6, 12, 18, 24, 30, 36, 42
The common multiples of 3, 4 and 6 are 12 and 24
So, the least common multiple of 3, 4 and 6 is 12.
We can find LCM of given numbers by listing multiples or by long division method.
3. Find the LCM of 18, 36 and 72 by division method.
Solution:
Write the numbers in a row separated by commas. Divide the numbers by a common prime number. We stop dividing after reaching the prime number. Find the product of divisors and the remainders.
So, LCM of 18, 36 and 72 is 2 × 3 × 3 × 1 × 2 × 4 = 432
Questions and Answers on Least Common Multiple:
I. Find the LCM of the given numbers. First one is shown for you as an example.
(i) 3 and 6
3 = 3, 6, 9, 12, 15, 18, 21, 24, 27 ………….
6 = 6, 12, 18, 24, 30, 36, 42 ………….
The common multiples of 3 and 6 are 6, 12, 18 ………….
Lowest common multiple of 3 and 6 is 6.
(ii) 2 and 4
(ii) 4 and 5
(iii) 3 and 12
(iv) 15 and 20
Answers:
I. (ii) 4
(ii20
(iii) 12
(iv) 60
You might like these
Common Multiples | How to Find Common Multiples of Two Numbers?
Common multiples of two or more given numbers are the numbers which can exactly be divided by each of the given numbers. Consider the following. (i) Multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, …………etc. Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, …………… etc.
Common Factors | Find the Common Factor | Worksheet | Answer
Common factors of two or more numbers are a number which divides each of the given numbers exactly. For examples 1. Find the common factor of 6 and 8. Factor of 6 = 1, 2, 3 and 6. Factor
Divisibility Rules | Divisibility Test|Divisibility Rules From 2 to 18
To find out factors of larger numbers quickly, we perform divisibility test. There are certain rules to check divisibility of numbers. Divisibility tests of a given number by any of the number 2, 3, 4, 5, 6, 7, 8, 9, 10 can be perform simply by examining the digits of the
Prime and Composite Numbers | Prime Numbers | Composite Numbers
What are the prime and composite numbers? Prime numbers are those numbers which have only two factors 1 and the number itself. Composite numbers are those numbers which have more than two factors.
Properties of Division | Division of Property Overview|Math Properties
The properties of division are discussed here: 1. If we divide a number by 1 the quotient is the number itself. In other words, when any number is divided by 1, we always get the number itself as the quotient. For example: (i) 7542 ÷ 1 = 7542 (ii) 372 ÷ 1 = 372
Multiplication by Ten, Hundred and Thousand |Multiply by 10, 100 &1000
To multiply a number by 10, 100, or 1000 we need to count the number of zeroes in the multiplier and write the same number of zeroes to the right of the multiplicand. Rules for the multiplication by 10, 100 and 1000: If we multiply a whole number by a 10, then we write one
Common Multiples.
Least Common Multiple (L.C.M).
To find Least Common Multiple by using Prime Factorization Method.
Examples to find Least Common Multiple by using Prime Factorization Method.
To Find Lowest Common Multiple by using Division Method
Examples to find Least Common Multiple of two numbers by using Division Method
Examples to find Least Common Multiple of three numbers by using Division Method
Relationship between H.C.F. and L.C.M.
Worksheet on H.C.F. and L.C.M.
Word problems on H.C.F. and L.C.M.
Worksheet on word problems on H.C.F. and L.C.M.
From Least Common Multiple to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Recent Articles
-
Dividing 3-Digit by 1-Digit Number | Long Division |Worksheet Answer
Apr 24, 24 03:46 PM
Dividing 3-Digit by 1-Digit Numbers are discussed here step-by-step. How to divide 3-digit numbers by single-digit numbers? Let us follow the examples to learn to divide 3-digit number by one-digit nu… -
Symmetrical Shapes | One, Two, Three, Four & Many-line Symmetry
Apr 24, 24 03:45 PM
Symmetrical shapes are discussed here in this topic. Any object or shape which can be cut in two equal halves in such a way that both the parts are exactly the same is called symmetrical. The line whi… -
Mental Math on Geometrical Shapes | Geometry Worksheets| Answer
Apr 24, 24 03:35 PM
In mental math on geometrical shapes we will solve different type of problems on simple closed curves, polygons, basic geometrical concepts, perpendicular lines, parallel lines, circle, terms relates… -
Circle Math | Terms Related to the Circle | Symbol of Circle O | Math
Apr 24, 24 02:57 PM
In circle math the terms related to the circle are discussed here. A circle is such a closed curve whose every point is equidistant from a fixed point called its centre. The symbol of circle is O. We… -
Fundamental Geometrical Concepts | Point | Line | Properties of Lines
Apr 24, 24 12:38 PM
The fundamental geometrical concepts depend on three basic concepts — point, line and plane. The terms cannot be precisely defined. However, the meanings of these terms are explained through examples.