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The Least Common Multiple (LCM) is also referred to as the Lowest Common Multiple (LCM) and Least Common Divisor (LCD). For two integers a and b, denoted LCM(a,b), the LCM is the smallest positive integer that is evenly divisible by both a and b. For example, LCM(2,3) = 6 and LCM(6,10) = 30. The LCM of two or more numbers is the smallest number that is evenly divisible by all numbers in the set. Least Common Multiple CalculatorFind the LCM of a set of numbers with this calculator which also shows the steps and how to do the work. Input the numbers you want to find the LCM for. You can use commas or spaces to separate your numbers. But do not use commas within your numbers. For example, enter 2500, 1000 and not 2,500, 1,000. How to Find the Least Common Multiple LCMThis LCM calculator with steps finds the LCM and shows the work using 6 different methods:
How to Find LCM by Listing Multiples
Example: LCM(6,7,21)
How to find LCM by Prime Factorization
The LCM(a,b) is calculated by finding the prime factorization of both a and b. Use the same process for the LCM of more than 2 numbers. For example, for LCM(12,30) we find:
For example, for LCM(24,300) we find:
How to find LCM by Prime Factorization using Exponents
Example: LCM(12,18,30)
Example: LCM(24,300)
How to Find LCM Using the Cake Method (Ladder Method)The cake method uses division to find the LCM of a set of numbers. People use the cake or ladder method as the fastest and easiest way to find the LCM because it is simple division. The cake method is the same as the ladder method, the box method, the factor box method and the grid method of shortcuts to find the LCM. The boxes and grids might look a little different, but they all use division by primes to find LCM. Find the LCM(10, 12, 15, 75)
How to Find the LCM Using the Division MethodFind the LCM(10, 18, 25)
How to Find LCM by GCFThe formula to find the LCM using the Greatest Common Factor GCF of a set of numbers is: LCM(a,b) = (a×b)/GCF(a,b) Example: Find LCM(6,10)
A factor is a number that results when you can evenly divide one number by another. In this sense, a factor is also known as a divisor. The greatest common factor of two or more numbers is the largest number shared by all the factors. The greatest common factor GCF is the same as:
How to Find the LCM Using Venn DiagramsVenn diagrams are drawn as overlapping circles. They are used to show common elements, or intersections, between 2 or more objects. In using Venn diagrams to find the LCM, prime factors of each number, we call the groups, are distributed among overlapping circles to show the intersections of the groups. Once the Venn diagram is completed you can find the LCM by finding the union of the elements shown in the diagram groups and multiplying them together. How to Find LCM of Decimal Numbers
Properties of LCMThe LCM is associative:LCM(a, b) = LCM(b, a) The LCM is commutative:LCM(a, b, c) = LCM(LCM(a, b), c) = LCM(a, LCM(b, c)) The LCM is distributive:LCM(da, db, dc) = dLCM(a, b, c) The LCM is related to the greatest common factor (GCF):LCM(a,b) = a × b / GCF(a,b) and GCF(a,b) = a × b / LCM(a,b) References[1] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition, New York, NY: CRC Press, 2003 p. 101. [2] Weisstein, Eric W. Least Common Multiple. From MathWorld--A Wolfram Web Resource.
We will learn the relationship between H.C.F. and L.C.M. of
two numbers. First we need to find the highest common factor (H.C.F.) of 15 and 18 which is 3. Then we need to find the lowest common multiple (L.C.M.) of 15 and 18 which is 90. H.C.F. × L.C.M. = 3 × 90 = 270 Also the product of numbers = 15 × 18 = 270 Therefore, product of H.C.F. and L.C.M. of 15 and 18 = product of 15 and 18. Again, let us consider the two numbers 16 and 24 Prime factors of 16 and 24 are: 16 = 2 × 2 × 2 × 2 24 = 2 × 2 × 2 × 3 L.C.M. of 16 and 24 is 48; H.C.F. of 16 and 24 is 8; L.C.M. × H.C.F. = 48 × 8 = 384 Product of numbers = 16 × 24 = 384 So, from the above explanations we conclude that the product of highest common factor (H.C.F.) and lowest common multiple (L.C.M.) of two numbers is equal to the product of two numbers or, H.C.F. × L.C.M. = First number × Second number or, L.C.M. = \(\frac{\textrm{First Number} \times \textrm{Second Number}}{\textrm{H.C.F.}}\) or, L.C.M. × H.C.F. = Product of two given numbers or, L.C.M. = \(\frac{\textrm{Product of Two Given Numbers}}{\textrm{H.C.F.}}\) or, H.C.F. = \(\frac{\textrm{Product of Two Given Numbers}}{\textrm{L.C.M.}}\) Solved examples on the
relationship between H.C.F. and L.C.M.: 1. Find the L.C.M. of 1683 and 1584. Solution: First we find highest common factor of 1683 and 1584 Therefore, highest common factor of 1683 and 1584 = 99 Lowest common multiple of 1683 and 1584 = First number × Second number/ H.C.F. = \(\frac{1584 × 1683}{99}\) = 26928 2. Highest common factor and lowest common multiple of two numbers are 18 and 1782 respectively. One number is 162, find the other. Solution: We know, H.C.F. × L.C.M. = First number × Second number then we get, 18 × 1782 = 162 × Second number \(\frac{18 × 1782}{162}\) = Second number Therefore, the second number = 198 3. The HCF of two numbers is 3 and their LCM is 54. If one of the numbers is 27, find the other number. Solution: HCF × LCM = Product of two numbers 3 × 54 = 27 × second number Second number = \(\frac{3 × 54}{27}\) Second number = 6 4. The highest common factor and the lowest common multiple of two numbers are 825 and 25 respectively. If one of the two numbers is 275, find the other number. Solution: We know, H.C.F. × L.C.M. = First number × Second number then we get, 825 × 25 = 275 × Second number \(\frac{825 × 25}{275}\) = Second number Therefore, the second number = 75 5. Find the H.C.F. and L.C.M. of 36 and 48. Solution:
Therefore, product of the two numbers = H.C.F × L.C.M. 2. The H.C.F. of two numbers 30 and 42 is 6. Find the L.C.M. Solution: We have H.C.F. × L.C.M. = product of the numbers 6 × L.C.M. = 30 × 42 L.C.M. = \[\frac{30 × 42}{\sqrt{6}}\] = \[\frac{1260}{\sqrt{6}}\] = 210 3. Find the greatest number which divides 105 and 180 completely. Solution:
Therefore, the greatest number that divides 105 and 180 completely is 15. 4. Find the least number which leaves 3 as remainder when divided by 24 and 42. Solution:
The least number which leaves 3 as remainder is 168 + 3 = 171. Important Notes: Two numbers which have only 1 as the common factor are called co-prime. The least common multiple (L.C.M.) of two or more numbers is the smallest number which is divisible by all the numbers. If two numbers are co-prime, their L.C.M. is the product of the numbers. If one number is the multiple of the other, then the multiple is their L.C.M. L.C.M. of two or more numbers cannot be less than any one of the given numbers. H.C.F. of two or more numbers is the highest number that can divide the numbers without leaving any remainder. If one number is a factor of the second number then the smaller number is the H.C.F. of the two given numbers. The product of L.C.M. and H.C.F. of two numbers is equal to the product of the two given numbers. Questions and Answers on Relationship between H.C.F. and L.C.M. 1. The H.C.F. of two numbers 20 and 75 is 5. Find their L.C.M. 2. The L.C.M. of two numbers 72 and 180 is 360. Find their H.C.F. 3. The L.C.M. of two numbers is 120. If the product of the numbers is 1440, find the H.C.F. 4. Find the least number which leaves 5 as remainder when divided by 36 and 54. 5. The product of two numbers is 384. If their H.C.F. is 8, find the L.C.M. Answer: 1. 300 2. 36 3. 12 4. 113 5. 48
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. 5th Grade Math Problems From Relationship between H.C.F. and L.C.M. to HOME PAGE
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