Lcm of 16 and 5

Lcm of 16 and 5 DEFAULT




What is the Least Common Multiple (LCM) of 5, 10, and 16? Here we will show you step-by-step how to find the Least Common Multiple of 5, 10, and 16.

Step 1)First we find and list the prime factors of 5, 10, and 16 (Prime Factorization):

Prime Factors of 5:
5

Prime Factors of 10:
2, 5

Prime Factors of 16:
2, 2, 2, 2

Step 2)Then we look at the frequency of the prime factors as they appear in each set above. List each prime factor the greatest number of times it occurs in any of the sets:

2, 2, 2, 2, 5

Step 3)Finally, we multiply the prime numbers from Step 2 together.

2 x 2 x 2 x 2 x 5 = 80

That's it. The Least Common Multiple (LCM) of 5, 10, and 16 is 80.


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LCM Calculator - Least Common Multiple

The LCM calculator will determine the least common multiple of two to fifteen numbers for you - no need to fret! This calculation is essential when adding or subtracting fractions with different denominators. The following text will explain what is LCM, show how to find the least common multiple and show how to use the least common multiple calculator.

What is LCM?

The LCM is the least common multiple or lowest common multiple between two or more numbers. We can find the least common multiple by breaking down each number into its prime factors. This can be accomplished by hand or by using the factor calculator or prime factorization calculator. The method for finding the LCM, along with an example illustrating the method, will be seen in the next section.

How to find the Least Common Multiple

Take each number and find its prime factors. Knowing various divisibility rules helps assist in this process.

  1. Any even number is divisible by .
  2. Any numbers whose sum of the digits is divisible by , is also divisible by
  3. A number is divisible by if the last two digits of the number form a number that is divisible by
  4. All numbers ending in or are divisible by .
  5. The number is divisible by if it is divisible by both and .
  6. A number is divisible by if the last three digits of the number form a number that is divisible by .
  7. A number whose digits sum to a number divisible by is also divisible by .
  8. Any number ending in is divisible by .

Once the numbers are broken down to their prime factors, multiply the highest power of each factor to get the LCM.

Least Common Multiple calculator

We are going to show how to find the LCM of , and . First we'll get the factors of each number. These are: ,, . Gather all the factors, so we have . Next multiply the highest power of each of these factors. That gives us . The LCM calculator can be used to check your answer or simply perform this calculation for you.

A Related Concept: The GCF

Just as you need prime factorization to get the LCM, it's equally important to find the GCF, which is the greatest common factor. To find the GCF, take the product of all the common factors of each number. For example, the GCF of and is since the only factor in common between the two numbers is . The GCF calculatoris a handy tool to calculate this.

Note that the LCM of two integers is the smallest positive integer the is divisible by both the integers. This is only true if both the integers are not zero. The LCM calculator will display a value of zero in such a case that one or more of the numbers is zero.

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Least common multiple

Smallest positive integer divisible by two or more integers

A Venn diagramshowing the least common multiples of combinations of 2, 3, 4, 5 and 7 (6 is skipped as it is 2 × 3, both of which are already represented).
For example, a card game which requires its cards to be divided equally among up to 5 players requires at least 60 cards, the number at the intersection of the 2, 3, 4, and 5 sets, but not the 7 set.

In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integersa and b, usually denoted by lcm(ab), is the smallest positive integer that is divisible by both a and b.[1][2] Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero.[3] However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility.

The lcm is the "lowest common denominator" (lcd) that can be used before fractions can be added, subtracted or compared. The lcm of more than two integers is also well-defined: it is the smallest positive integer that is divisible by each of them.[1]

Overview[edit]

A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and −2 as well.

Notation[edit]

The least common multiple of two integers a and b is denoted as lcm(a, b).[1] Some older textbooks use [a, b].[3][4]

Example[edit]

{\displaystyle \operatorname {lcm} (4,6)}

Multiples of 4 are:

{\displaystyle 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,...}

Multiples of 6 are:

{\displaystyle 6,12,18,24,30,36,42,48,54,60,66,72,...}

Common multiples of 4 and 6 are the numbers that are in both lists:

{\displaystyle 12,24,36,48,60,72,...}

In this list, the smallest number is 12. Hence, the least common multiple is 12.

Applications[edit]

When adding, subtracting, or comparing simple fractions, the least common multiple of the denominators (often called the lowest common denominator) is used, because each of the fractions can be expressed as a fraction with this denominator. For example,

{2\over21}+{1\over6}={4\over42}+{7\over42}={11\over42}

where the denominator 42 was used, because it is the least common multiple of 21 and 6.

Gears problem[edit]

Suppose there are two meshing gears in a machine, having m and n teeth, respectively, and the gears are marked by a line segment drawn from the center of the first gear to the center of the second gear. When the gears begin rotating, the number of rotations the first gear must complete to realign the line segment can be calculated by using {\displaystyle \operatorname {lcm} (m,n)}. The first gear must complete {\displaystyle \operatorname {lcm} (m,n) \over m} rotations for the realignment. By that time, the second gear will have made {\displaystyle \operatorname {lcm} (m,n) \over n} rotations.

Planetary alignment[edit]

See also: Planetary alignment

Suppose there are three planets revolving around a star which take l, m and n units of time, respectively, to complete their orbits. Assume that l, m and n are integers. Assuming the planets started moving around the star after an initial linear alignment, all the planets attain a linear alignment again after {\displaystyle \operatorname {lcm} (l,m,n)} units of time. At this time, the first, second and third planet will have completed {\displaystyle \operatorname {lcm} (l,m,n) \over l}, {\displaystyle \operatorname {lcm} (l,m,n) \over m} and {\displaystyle \operatorname {lcm} (l,m,n) \over n} orbits, respectively, around the star.[5]

Calculation[edit]

Using the greatest common divisor[edit]

The following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor (gcd), also known as the greatest common factor:

{\displaystyle \operatorname {lcm} (a,b)={\frac {|ab|}{\gcd(a,b)}}.}

This formula is also valid when exactly one of a and b is 0, since gcd(a, 0) = |a|. However, if both a and b are 0, this formula would cause division by zero; lcm(0, 0) = 0 is a special case.

There are fast algorithms for computing the gcd that do not require the numbers to be factored, such as the Euclidean algorithm. To return to the example above,

{\displaystyle \operatorname {lcm} (21,6)={21\cdot 6 \over \gcd(21,6)}={21\cdot 6 \over \gcd(3,6)}={21\cdot 6 \over 3}={\frac {126}{3}}=42.}

Because gcd(a, b) is a divisor of both a and b, it is more efficient to compute the lcm by dividing before multiplying:

{\displaystyle \operatorname {lcm} (a,b)=\left({|a| \over \gcd(a,b)}\right)\cdot |b|=\left({|b| \over \gcd(a,b)}\right)\cdot |a|.}

This reduces the size of one input for both the division and the multiplication, and reduces the required storage needed for intermediate results (that is, overflow in the a×b computation). Because gcd(a, b) is a divisor of both a and b, the division is guaranteed to yield an integer, so the intermediate result can be stored in an integer. Implemented this way, the previous example becomes:

{\displaystyle \operatorname {lcm} (21,6)={21 \over \gcd(21,6)}\cdot 6={21 \over \gcd(3,6)}\cdot 6={21 \over 3}\cdot 6=7\cdot 6=42.}

Using prime factorization[edit]

The unique factorization theorem indicates that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined, make up a composite number.

For example:

{\displaystyle 90=2^{1}\cdot 3^{2}\cdot 5^{1}=2\cdot 3\cdot 3\cdot 5.}

Here, the composite number 90 is made up of one atom of the prime number 2, two atoms of the prime number 3, and one atom of the prime number 5.

This fact can be used to find the lcm of a set of numbers.

Example: lcm(8,9,21)

Factor each number and express it as a product of prime number powers.

{\displaystyle {\begin{aligned}8&=2^{3}\\9&=3^{2}\\21&=3^{1}\cdot 7^{1}\end{aligned}}}

The lcm will be the product of multiplying the highest power of each prime number together. The highest power of the three prime numbers 2, 3, and 7 is 23, 32, and 71, respectively. Thus,

{\displaystyle \operatorname {lcm} (8,9,21)=2^{3}\cdot 3^{2}\cdot 7^{1}=8\cdot 9\cdot 7=504.}

This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization.

The same method can also be illustrated with a Venn diagram as follows, with the prime factorization of each of the two numbers demonstrated in each circle and all factors they share in common in the intersection. The lcm then can be found by multiplying all of the prime numbers in the diagram.

Here is an example:

48 = 2 × 2 × 2 × 2 × 3,
180 = 2 × 2 × 3 × 3 × 5,

sharing two "2"s and a "3" in common:

Least common multiple.svg
Least common multiple = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720
Greatest common divisor = 2 × 2 × 3 = 12

This also works for the greatest common divisor (gcd), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the gcd of 48 and 180 is 2 × 2 × 3 = 12.

Using a simple algorithm[edit]

This method works easily for finding the lcm of several integers.[citation needed]

Let there be a finite sequence of positive integers X = (x1, x2, ..., xn), n > 1. The algorithm proceeds in steps as follows: on each step m it examines and updates the sequence X(m) = (x1(m), x2(m), ..., xn(m)), X(1) = X, where X(m) is the mth iteration of X, that is, X at step m of the algorithm, etc. The purpose of the examination is to pick the least (perhaps, one of many) element of the sequence X(m). Assuming xk0(m) is the selected element, the sequence X(m+1) is defined as

xk(m+1) = xk(m), kk0
xk0(m+1) = xk0(m) + xk0(1).

In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X(m) to X(m+1) unchanged.

The algorithm stops when all elements in sequence X(m) are equal. Their common value L is exactly lcm(X).

For example, if X = X(1) = (3, 4, 6), the steps in the algorithm produce:

X(2) = (6, 4, 6)
X(3) = (6, 8, 6)
X(4) = (6, 8, 12) - by choosing the second 6
X(5) = (9, 8, 12)
X(6) = (9, 12, 12)
X(7) = (12, 12, 12) so lcm = 12.

Using the table-method[edit]

This method works for any number of numbers. One begins by listing all of the numbers vertically in a table (in this example 4, 7, 12, 21, and 42):

The process begins by dividing all of the numbers by 2. If 2 divides any of them evenly, write 2 in a new column at the top of the table, and the result of division by 2 of each number in the space to the right in this new column. If a number is not evenly divisible, just rewrite the number again. If 2 does not divide evenly into any of the numbers, repeat this procedure with the next largest prime number, 3 (see below).

× 2
4 2
7 7
12 6
21 21
42 21

Now, assuming that 2 did divide at least one number (as in this example), check if 2 divides again:

× 2 2
4 21
7 7 7
12 63
21 21 21
42 2121

Once 2 no longer divides any number in the current column, repeat the procedure by dividing by the next larger prime, 3. Once 3 no longer divides, try the next larger primes, 5 then 7, etc. The process ends when all of the numbers have been reduced to 1 (the column under the last prime divisor consists only of 1's).

× 2 2 3 7
4 211 1
7 7 7 7 1
12 6311
21 21 21 71
42 2121 71

Now, multiply the numbers in the top row to obtain the lcm. In this case, it is 2 × 2 × 3 × 7 = 84.

As a general computational algorithm, the above is quite inefficient. One would never want to implement it in software: it takes too many steps and requires too much storage space. A far more efficient numerical algorithm can be obtained by using Euclid's algorithm to compute the gcd first, and then obtaining the lcm by division.

Formulas[edit]

Fundamental theorem of arithmetic[edit]

According to the fundamental theorem of arithmetic, a positive integer is the product of prime numbers, and this representation is unique up to the ordering of prime numbers:

{\displaystyle n=2^{n_{2}}3^{n_{3}}5^{n_{5}}7^{n_{7}}\cdots =\prod _{p}p^{n_{p}},}

where the exponents n2, n3, ... are non-negative integers; for example, 84 = 22 31 50 71 110 130 ...

Given two positive integers {\textstyle a=\prod _{p}p^{a_{p}}} and {\textstyle b=\prod _{p}p^{b_{p}}}, their least common multiple and greatest common divisor are given by the formulas

{\displaystyle \gcd(a,b)=\prod _{p}p^{\min(a_{p},b_{p})}}

and

{\displaystyle \operatorname {lcm} (a,b)=\prod _{p}p^{\max(a_{p},b_{p})}.}

Since

{\displaystyle \min(x,y)+\max(x,y)=x+y,}

this gives

{\displaystyle \gcd(a,b)\operatorname {lcm} (a,b)=ab.}

In fact, every rational number can be written uniquely as the product of primes, if negative exponents are allowed. When this is done, the above formulas remain valid. For example:

{\displaystyle {\begin{aligned}4&=2^{2}3^{0},&6&=2^{1}3^{1},&\gcd(4,6)&=2^{1}3^{0}=2,&\operatorname {lcm} (4,6)&=2^{2}3^{1}=12.\\[8pt]{\tfrac {1}{3}}&=2^{0}3^{-1}5^{0},&{\tfrac {2}{5}}&=2^{1}3^{0}5^{-1},&\gcd \left({\tfrac {1}{3}},{\tfrac {2}{5}}\right)&=2^{0}3^{-1}5^{-1}={\tfrac {1}{15}},&\operatorname {lcm} \left({\tfrac {1}{3}},{\tfrac {2}{5}}\right)&=2^{1}3^{0}5^{0}=2,\\[8pt]{\tfrac {1}{6}}&=2^{-1}3^{-1},&{\tfrac {3}{4}}&=2^{-2}3^{1},&\gcd \left({\tfrac {1}{6}},{\tfrac {3}{4}}\right)&=2^{-2}3^{-1}={\tfrac {1}{12}},&\operatorname {lcm} \left({\tfrac {1}{6}},{\tfrac {3}{4}}\right)&=2^{-1}3^{1}={\tfrac {3}{2}}.\end{aligned}}}

Lattice-theoretic[edit]

The positive integers may be partially ordered by divisibility: if a divides b (that is, if b is an integer multiple of a) write ab (or equivalently, ba). (Note that the usual magnitude-based definition of ≤ is not used here.)

Under this ordering, the positive integers become a lattice, with meet given by the gcd and join given by the lcm. The proof is straightforward, if a bit tedious; it amounts to checking that lcm and gcd satisfy the axioms for meet and join. Putting the lcm and gcd into this more general context establishes a duality between them:

If a formula involving integer variables, gcd, lcm, ≤ and ≥ is true, then the formula obtained by switching gcd with lcm and switching ≥ with ≤ is also true. (Remember ≤ is defined as divides).

The following pairs of dual formulas are special cases of general lattice-theoretic identities.

Commutative laws
{\displaystyle \operatorname {lcm} (a,b)=\operatorname {lcm} (b,a),}
{\displaystyle \gcd(a,b)=\gcd(b,a).}
    
Associative laws
{\displaystyle \operatorname {lcm} (a,\operatorname {lcm} (b,c))=\operatorname {lcm} (\operatorname {lcm} (a,b),c),}
{\displaystyle \gcd(a,\gcd(b,c))=\gcd(\gcd(a,b),c).}
    
Absorption laws
{\displaystyle \operatorname {lcm} (a,\gcd(a,b))=a,}
{\displaystyle \gcd(a,\operatorname {lcm} (a,b))=a.}
Idempotent laws
{\displaystyle \operatorname {lcm} (a,a)=a,}
{\displaystyle \gcd(a,a)=a.}
    
Define divides in terms of lcm and gcd
{\displaystyle a\geq b\iff a=\operatorname {lcm} (a,b),}
{\displaystyle a\leq b\iff a=\gcd(a,b).}

It can also be shown[6] that this lattice is distributive; that is, lcm distributes over gcd and gcd distributes over lcm:

{\displaystyle \operatorname {lcm} (a,\gcd(b,c))=\gcd(\operatorname {lcm} (a,b),\operatorname {lcm} (a,c)),}
{\displaystyle \gcd(a,\operatorname {lcm} (b,c))=\operatorname {lcm} (\gcd(a,b),\gcd(a,c)).}

This identity is self-dual:

{\displaystyle \gcd(\operatorname {lcm} (a,b),\operatorname {lcm} (b,c),\operatorname {lcm} (a,c))=\operatorname {lcm} (\gcd(a,b),\gcd(b,c),\gcd(a,c)).}

Other[edit]

  • Let D be the product of ω(D) distinct prime numbers (that is, D is squarefree).

Then[7]

{\displaystyle |\{(x,y)\;:\;\operatorname {lcm} (x,y)=D\}|=3^{\omega (D)},}

where the absolute bars || denote the cardinality of a set.

  • If none of {\displaystyle a_{1},a_{2},\ldots ,a_{r}} is zero, then
{\displaystyle \operatorname {lcm} (a_{1},a_{2},\ldots ,a_{r})=\operatorname {lcm} (\operatorname {lcm} (a_{1},a_{2},\ldots ,a_{r-1}),a_{r}).}[8][9]

In commutative rings[edit]

The least common multiple can be defined generally over commutative rings as follows: Let a and b be elements of a commutative ring R. A common multiple of a and b is an element m of R such that both a and b divide m (that is, there exist elements x and y of R such that ax = m and by = m). A least common multiple of a and b is a common multiple that is minimal, in the sense that for any other common multiple n of a and b, m divides n.

In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are associates. In a unique factorization domain, any two elements have a least common multiple. In a principal ideal domain, the least common multiple of a and b can be characterised as a generator of the intersection of the ideals generated by a and b (the intersection of a collection of ideals is always an ideal).

See also[edit]

Notes[edit]

  1. ^ abcWeisstein, Eric W. "Least Common Multiple". mathworld.wolfram.com. Retrieved 2020-08-30.
  2. ^Hardy & Wright, § 5.1, p. 48
  3. ^ abLong (1972, p. 39)
  4. ^Pettofrezzo & Byrkit (1970, p. 56)
  5. ^"nasa spacemath"(PDF).
  6. ^The next three formulas are from Landau, Ex. III.3, p. 254
  7. ^Crandall & Pomerance, ex. 2.4, p. 101.
  8. ^Long (1972, p. 41)
  9. ^Pettofrezzo & Byrkit (1970, p. 58)

References[edit]

  • Crandall, Richard; Pomerance, Carl (2001), Prime Numbers: A Computational Perspective, New York: Springer, ISBN 
  • Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 
  • Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea
  • Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950
  • Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766
Sours: https://en.wikipedia.org/wiki/Least_common_multiple
GCF

LCM (Lowest Common Multiple)

LCM Calculator

Answers to Questions (FAQ)

How to calculate the LCM? (Algorithm)

Method 1: list all multiples and find the .

Example: for and
has for multiples
has for multiples
The is .

Method 2: use the prime factors decomposition. The is the multiplication of common factors by non-common factors

Example: $ 10 = 2 \times 5 $ and $ 12 = 2 \times 2 \times 3 $
Common factors: and non common factors:
LCM(10, 12) = $ 2 \times 2 \times 3 \times 5 = 60 $

Method 3: use the GCD value and apply the formula

Example: GCD(10, 12) = 2
LCM(10, 12) = (10 * 12) / 2 = 60

How to calculate the LCM with multiple numbers? (LCM of 2 numbers or more)

Method 1: list all multiples and find the .

Example: for 10, 12 and 15
10 has for multiples 0,10,20,30,40,50,60,70 etc.
12 has for multiples 0,12,24,36,48,60,72 etc.
15 has for multiples 0,15,30,45,60,75 etc.
The is 60.

Method 2: apply the by 2 and use the formula

Example: LCM(10, 12) = 60
LCM(10, 12, 15) = ( LCM(10, 12) , 15 ) = LCM(60,15) = 60

How to calculate the lowest common denominator of fractions?

To calculate fractions and/or set fractions with the same denominator, calculate the of the denominators (the fraction below the fraction line).

Example: The fractions 7/8 and 15/36, their smallest common denominator is LCM(8,36)=72.
7/8 can therefore be written as 63/72 and 15/36 can be written 30/72.

How to calculate LCM with a calculator (TI or Casio)?

Calculators has generally a function for , else with GCD function, apply the formula:

$$ \text{L C M}(a, b) = \frac{ a \times b} { \text{G C D}(a, b) } $$

How to calculate LCM with a zero 0?

0 has no multiple, because no number can be divided by zero

How to calculate LCM with non-integers?

as it is mathematically defined, has no sense with non integers. However, it is possible to use this formula: CM(a*c,b*c) = CM(a,b)*c where CM is a common multiple (not the lowest) other rational numbers.

Example: CM(1.2,2.4) = CM(12,24)/10 = 2

What are LCM for the N first integers?

The following numbers have the property of having many divisors, some of them are highly composite numbers.

LCM(1,2,3)=6
LCM(1,2,3,4)=12
LCM(1,2,3,4,5)=60
LCM(1,2,3,4,5,6)=60
LCM(1,2,3...6,7)=420
LCM(1,2,3...7,8)=840
LCM(1,2,3...8,9)=2520
LCM(1,2,3...9,10)=2520
LCM(1,2,3...10,11)=27720
LCM(1,2,3...11,12)=27720
LCM(1,2,3...12,13)=360360
LCM(1,2,3...13,14)=360360
LCM(1,2,3...14,15)=360360
LCM(1,2,3...15,16)=720720
LCM(1,2,3...16,17)=12252240
LCM(1,2,3...17,18)=12252240
LCM(1,2,3...18,19)=232792560
LCM(1,2,3...19,20)=232792560
LCM(1,2,3...20,21)=232792560
LCM(1,2,3...21,22)=232792560
LCM(1,2,3...22,23)=5354228880
LCM(1,2,3...23,24)=5354228880
LCM(1,2,3...24,25)=26771144400
LCM(1,2,3...25,26)=26771144400
LCM(1,2,3...26,27)=80313433200
LCM(1,2,3...27,28)=80313433200
LCM(1,2,3...28,29)=2329089562800
LCM(1,2,3...29,30)=2329089562800
LCM(1,2,3...30,31)=72201776446800
LCM(1,2,3...31,32)=144403552893600
LCM(1,2,3...32,33)=144403552893600
LCM(1,2,3...33,34)=144403552893600
LCM(1,2,3...34,35)=144403552893600
LCM(1,2,3...35,36)=144403552893600
LCM(1,2,3...36,37)=5342931457063200
LCM(1,2,3...37,38)=5342931457063200
LCM(1,2,3...38,39)=5342931457063200
LCM(1,2,3...39,40)=5342931457063200
LCM(1,2,3...40,41)=219060189739591200
LCM(1,2,3...41,42)=219060189739591200
LCM(1,2,3...42,43)=9419588158802421600
LCM(1,2,3...43,44)=9419588158802421600
LCM(1,2,3...44,45)=9419588158802421600
LCM(1,2,3...45,46)=9419588158802421600
LCM(1,2,3...46,47)=442720643463713815200
LCM(1,2,3...47,48)=442720643463713815200
LCM(1,2,3...48,49)=3099044504245996706400

Why the LCM of 2 consecutive numbers is a multiple of 2?

For any couple of 2 consecutive numbers, one is even and the other is odd, so only one is a multiple of 2. According to the method of computation of the via the decomposition in prime factors, then the is necessarily multiple of 2 which is a not common factor for the 2 numbers.

Why the LCM of 3 consecutive numbers is a multiple of 3?

For any triplet of 3 consecutive numbers, only one is multiple of 3. According to the method of computation of the via the decomposition in prime factors, then the is necessarily multiple of 3 which is a not common factor for the 3 numbers.

What is the difference between LCM and GCD?

The is a common multiple of the 2 numbers, which is therefore a larger number having for divider the 2 numbers.

The GCD is a common divisor of the 2 numbers, which is therefore a smaller number having for multiple the 2 numbers.

The and the CGD are linked by the formula: $$ \ text {L C M} (a, b) = \ frac {a \ times b} {\ text {G C D} (a, b)} $$

Why calculate the LCM?

PPCM is a number that is a multiple of many, and it's as small as possible. This gives it a lot of mathematical advantage and simplifies the calculations.

Example: A circle has 360° because 360 is divisible by 1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360 which is very practical.

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5 and of lcm 16

Least Common Multiple

The smallest positive number that is a multiple of two or more numbers.

Let's start with an Example ... 

Least Common Multiple of 3 and 5:

List the Multiples of each number,

The multiples of 3 are 3, 6, 9, 12, 15, 18, ... etc
The multiples of 5 are 5, 10, 15, 20, 25, ... etc

Find the first Common (same) value:

LCM of 3 and 5 is 15

The Least Common Multiple of 3 and 5 is 15

(15 is a multiple of both 3 and 5, and is the smallest number like that.)

So ... what is a "Multiple" ?

We get a multiple of a number when we multiply it by another number. Such as multiplying by 1, 2, 3, 4, 5, etc, but not zero. Just like the multiplication table.

Here are some examples:

The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,...
The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...

What is a "Common Multiple" ?

Say we have listed the first few multiples of 4 and 5: the common multiples are those that are found in both lists:

The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,...
The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...

Notice that 20 and 40 appear in both lists?
So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too)

What is the "Least Common Multiple" ?

It is simply the smallest of the common multiples.

In our previous example, the smallest of the common multiples is 20 ...

... so the Least Common Multiple of 4 and 5 is 20.

Finding the Least Common Multiple

List the multiples of the numbers until we get our first match.

Example: Find the least common multiple of 4 and 10:

The multiples of 4 are: 4, 8, 12, 16, 20, ...
and the multiples of 10 are: 10, 20, ...

Aha! there is a match at 20. It looks like this:

LCM of 4 and 10 is 20

So the least common multiple of 4 and 10 is 20

Example: Find the least common multiple of 6 and 15:

The multiples of 6 are: 6, 12, 18, 24, 30, ...
and the multiples of 15 are: 15, 30, ...

There is a match at 30

So the least common multiple of 6 and 15 is 30

More than 2 Numbers

We can also find the least common multiple of three (or more) numbers.

Example: Find the least common multiple of 4, 6, and 8

Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Multiples of 6 are: 6, 12, 18, 24, 30, 36, ...
Multiples of 8 are: 8, 16, 24, 32, 40, ....

So 24 is the least common multiple (I can't find a smaller one!)

Hint: We can have smaller lists for the bigger numbers.

Least Common Multiple Tool

There is another method: the Least Common Multiple Tool does it automatically. (Yes, we waited until the end to tell you!)

 

 

Introduction to FractionsSimplifying FractionsComparing FractionsLeast Common Multiple ToolLeast Common DenominatorGreatest Common FactorAdding FractionsSubtracting FractionsFractions Index

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Sours: https://www.mathsisfun.com/least-common-multiple.html
Least Common Multiple (LCM) - Mathematics Grade 4 - Periwinkle

LCM of 5 and 16

FAQs on LCM of 5 and 16

What is the LCM of 5 and 16?

The LCM of 5 and 16 is 80. To find the least common multiple of 5 and 16, we need to find the multiples of 5 and 16 (multiples of 5 = 5, 10, 15, 20 . . . . 80; multiples of 16 = 16, 32, 48, 64 . . . . 80) and choose the smallest multiple that is exactly divisible by 5 and 16, i.e., 80.

What is the Relation Between GCF and LCM of 5, 16?

The following equation can be used to express the relation between GCF and LCM of 5 and 16, i.e. GCF × LCM = 5 × 16.

How to Find the LCM of 5 and 16 by Prime Factorization?

To find the LCM of 5 and 16 using prime factorization, we will find the prime factors, (5 = 5) and (16 = 2 × 2 × 2 × 2). LCM of 5 and 16 is the product of prime factors raised to their respective highest exponent among the numbers 5 and 16.
⇒ LCM of 5, 16 = 24 × 51 = 80.

If the LCM of 16 and 5 is 80, Find its GCF.

LCM(16, 5) × GCF(16, 5) = 16 × 5
Since the LCM of 16 and 5 = 80
⇒ 80 × GCF(16, 5) = 80
Therefore, the GCF (greatest common factor) = 80/80 = 1.

What are the Methods to Find LCM of 5 and 16?

The commonly used methods to find the LCM of 5 and 16 are:

  • Prime Factorization Method
  • Division Method
  • Listing Multiples

 

Sours: https://www.cuemath.com/numbers/lcm-of-5-and-16/

Now discussing:

Least Common Multiple Calculator

Please provide numbers separated by a comma "," and click the "Calculate" button to find the LCM.



What is the Least Common Multiple (LCM)?

In mathematics, the least common multiple, also known as the lowest common multiple of two (or more) integers a and b, is the smallest positive integer that is divisible by both. It is commonly denoted as LCM(a, b).

Brute Force Method

There are multiple ways to find a least common multiple. The most basic is simply using a "brute force" method that lists out each integer's multiples.

EX:   Find LCM(18, 26)
18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234
26: 52, 78, 104, 130, 156, 182, 208, 234

As can be seen, this method can be fairly tedious, and is far from ideal.

Prime Factorization Method

A more systematic way to find the LCM of some given integers is to use prime factorization. Prime factorization involves breaking down each of the numbers being compared into its product of prime numbers. The LCM is then determined by multiplying the highest power of each prime number together. Note that computing the LCM this way, while more efficient than using the "brute force" method, is still limited to smaller numbers. Refer to the example below for clarification on how to use prime factorization to determine the LCM:

EX:   Find LCM(21, 14, 38)
21 = 3 × 7
14 = 2 × 7
38 = 2 × 19

The LCM is therefore:
3 × 7 × 2 × 19 = 798

Greatest Common Divisor Method

A third viable method for finding the LCM of some given integers is using the greatest common divisor. This is also frequently referred to as the greatest common factor (GCF), among other names. Refer to the link for details on how to determine the greatest common divisor. Given LCM(a, b), the procedure for finding the LCM using GCF is to divide the product of the numbers a and b by their GCF, i.e. (a × b)/GCF(a,b). When trying to determine the LCM of more than two numbers, for example LCM(a, b, c) find the LCM of a and b where the result will be q. Then find the LCM of c and q. The result will be the LCM of all three numbers. Using the previous example:

EX:   Find LCM(21, 14, 38)

GCF(14, 38) = 2
LCM(14, 38) =   = 266

GCF(266, 21) = 7
LCM(266, 21) =   = 798

LCM(21, 14, 38) = 798

Note that it is not important which LCM is calculated first as long as all the numbers are used, and the method is followed accurately. Depending on the particular situation, each method has its own merits, and the user can decide which method to pursue at their own discretion.

Sours: https://www.calculator.net/lcm-calculator.html


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