Question:

# What is the least common multiple of 16 28 20?

## 560 is the least common multiple.

The RABe 514 is a four-car double decker electrical multiple unit used by the Swiss Federal Railways SBB-CFF-FFS for the Zürich S-Bahn. It is part of the Siemens Desiro Double Deck product family. The trains are also referred to as DTZ which stands for the German word Doppelstocktriebzug (English: double decker multiple unit). On 23 February 2003 the Swiss Federal Railways' board of directors decided to give the 447 million CHF contract for building 35 double decker trains to Siemens Transportation Systems. This decision came as a surprise since Siemens had never built double decker EMUs before, except for an experimental train built in a consortium with DWA Görlitz (now Bombardier Transportation) that never entered into service. To fulfill the domestic content requirement in the contact, Siemens reached an agreement with Stadler Rail to perform some of the assembling and the commissioning in their factory in Altenrhein, Switzerland. The trains were originally intended to enter service in December 2005, but the date could not be kept. Nevertheless, the first trainset was presented to the public on 2 December 2005 at Zürich Hauptbahnhof. Until May of the next year, the RABe 514 were thoroughly tested and then entered into regular passenger service on the S14 line. In March 2006 the Swiss Federal Railways exercised their purchase option for another 25 units. Because of the delayed delivery of the first trainsets, Siemens agreed to build an additional train instead of paying a penalty. Delivery of all 61 trains was completed in July 2009. The DTZ trains are the second generation double decker trains used on the Zürich S-Bahn. Compared to the Re 450, the first generation trains, the RABe 514 feature a low-level entrance for level boarding, air conditioning and vacuum toilets (in a washroom suitable for the disabled). The four-car multiple unit consists of two powered end cars with two unpowered cars between them. Both bogies in an end car are driven by induction motors with a power output of 400 kW per axle providing a total of 3200 kW for the trainset. Since there was not enough space for a 15 kV power line through the train, both end cars draw their power from a separate pantograph. An automatic coupling system allows for up to four trainset to be connected together for additional capacity, however in practice the maximum is three connected trains due to the limited platform lengths of 300 m at the train stations. The double decker trains provide 74 seats in first class, 304 seats in second class as well room for about 600 people standing. The first class seats are equipped with normal 230 V power outlets for charging notebooks and other devices. The RABe 514 are used on lines previously served by Re 450. The thereby freed up Re 450 trains were then used to replace the old single deck trains RABDe 510 and RBe 540 from the 1960s. The first line served by the new Siemens trains was the S14 (Hinwil–Zürich Hauptbahnhof). Starting in December 2006 the S7 (Rapperswil–Zürich–Winterthur) line was also equipped with the new trains. However all newly delivered trains were first put on the S14 line because that line is better suited for testing due to its short length and lower importance than other lines. After the S7 line got all of its required 15 trainsets, the next trains were intended to go to the S5 line, but that plan was abandoned in favor of the S15 (Affoltern am Albis-Zürich-Rapperswil). With delivery of the second batch, the S8 (Pfäffikon-Zürich-Winterthur) line was equipped as well. On some of the S16 (Meilen-Zürich-Thayngen) services the DTZ trains are also used, especially on weekends. The original plan to couple first and second generation trains together to provide level-boarding on more lines could not be implemented due to software problems when connecting the two train generations.
NSB Class 69 (Norwegian: ) is an electric multiple unit used by Norges Statsbaner for a variety of commuter trains on the Norwegian railway system, as well as a few medium distance and branch line trains. It is the most common type of trainset in Norway, although the newer NSB type 72 has also been introduced. All the trains were built by Strømmen. During the 1960s NSB realized that they would need a new generation of electric multiple units for local traffic. Both the Class 65, 67 and 68 had for thirty years been built with slight modifications, and NSB needed both new and more modern trains for their operations. Among the inspiration was the successful X1 units used in Sweden. NSB decided on a number of rationalizations, first of all a new interior so that two new cars could hold the same capacity as three units from the older models. Secondly NSB wanted quicker trains, and increased the maximum speed from 70 to 130 km/h (81 mph). This increase in speed was sufficient to reduce the number of trains for a given frequency by a third. For instance on the line from Oslo Ø to Ski this allowed NSB to reduce the number of operative car from nine to four. The 69-set was also given new thyristor engines with 1,200 kW, a lot more than the old units. The first units were delivered on 1 November 1970, and the first series fifteen units (A-series) was put into service from Oslo Ø to Lillestrøm and Ski. The B-series was delivered a few years later, and used for longer lines, soon followed by more C-series units. The D-series was delivered in the 1980s and custom made for longer distances. The last series was delivered in 1993 as part of the stock for the 1994 Winter Olympics. All the original units consisted of twin-car sets, but from 1987 NSB ordered additional middle cars, to make three-car units. This allowed somewhat more flexibility, and NSB kept the A- and B-series as twin-cars so they could combine trains to make any number of cars needed between two and nine. The second two batches of D-series trains delivered in 1990 and 1993 were delivered with three cars. Not until 1982 were the 69-units put into service outside the Oslo area. At first they were tried out on Flåm Line, then in 1984 on the Bergen Commuter Rail. From 1991 they were also used on the Stavanger Commuter Rail. The E- and G-series are rebuilt sets, with higher comfort levels, the former in 1994 for Sørland Line between Kristiansand and Stavanger, the latter on Gjøvik Line in 2005. On 14 December 1999 NSB introduced wrap advertising for Freia Melkesjokolade on three of the trains, but chose to discontinue outside advertisements after a while. The BM69 trainsets are in use between Kongsberg–Eidsvoll, Spikkestad–Moss, Drammen–Dal, (Drammen)–Asker–Lillestrøm, (Skøyen)–Oslo S–Gjøvik, Skøyen–Ski, Skøyen–Mysen, and Skøyen–Årnes. The BM69 comes in six series: No. 04, 17, 20, 21, 22, 23.
In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. If either a or b is 0, LCM(ab) is defined to be zero. The LCM is familiar from grade-school arithmetic as the "least common denominator" (LCD) that must be determined before fractions can be added, subtracted or compared. The LCM of more than two integers is also well-defined: it is the smallest integer that is divisible by each of them. 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. In this article we will denote the least common multiple of two integers a and b as lcm( a, b ). Some older textbooks use [ a, b ]. What is the LCM of 4 and 6? Multiples of 4 are: and the multiples of 6 are: Common multiples of 4 and 6 are simply the numbers that are in both lists: So, from this list of the first few common multiples of the numbers 4 and 6, their least common multiple is 12. When adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. For instance, where the denominator 42 was used because it is the least common multiple of 21 and 6. Many school age children are taught the term greatest common factor (GCF) instead of the greatest common divisor(GCD); therefore, for those familiar with the concept of GCF, substitute GCF when GCD is used below. The following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor (GCD): This formula is also valid when exactly one of a and b is 0, since gcd(a, 0) = |a|. 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, Because gcd(a, b) is a divisor of both a and b, it's more efficient to compute the LCM by dividing before multiplying: This reduces the size of one input for both the division and the multiplication, and reduces the required storage needed for intermediate results (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. Done this way, the previous example becomes: The unique factorization theorem says 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 together, make up a composite number. For example: Here we have the composite number 90 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 knowledge can be used to find the LCM of a set of numbers. Example: Find the value of lcm(8,9,21). First, factor out each number and express it as a product of prime number powers. 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, This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization, but is useful for illustrating concepts. This method can be illustrated using a Venn diagram as follows. Find the prime factorization of each of the two numbers. Put the prime factors into a Venn diagram with one circle for each of the two numbers, and all factors they share in common in the intersection. To find the LCM, just multiply all of the prime numbers in the diagram. Here is an example: and what they share in common is two "2"s and a "3": 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. This method works as easily for finding the LCM of several integers. 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, i.e. X at step m of the algorithm, etc. The purpose of the examination is to pick up the least (perhaps, one of many) element of the sequence X)m(. Assuming xk0)m( is the selected element, the sequence X+1)m( is defined as In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X)m( to X+1)m( unchanged. The algorithm stops when all elements in sequence X)m( are equal. Their common value L is exactly LCM(X). (For a proof and an interactive simulation see reference below, Algorithm for Computing the LCM.) This method works for any number of factors. 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 factors by 2. If any of them divides evenly, write 2 at the top of the table and the result of division by 2 of each factor in the space to the right of each factor and below the 2. If a number does not divide evenly, just rewrite the number again. If 2 does not divide evenly into any of the numbers, try 3. Now, check if 2 divides again: Once 2 no longer divides, divide by 3. If 3 no longer divides, try 5 and 7. Keep going until all of the numbers have been reduced to 1. Now, multiply the numbers on the top and you have the LCM. In this case, it is 2 × 2 × 3 × 7 = 84. You will get to the LCM the quickest if you use prime numbers and start from the lowest prime, 2. According to the fundamental theorem of arithmetic a positive integer is the product of prime numbers, and, except for their order, this representation is unique: where the exponents n2, n3, ... are non-negative integers; for example, 84 = 22 31 50 71 110 130 ... Given two integers   $\;a=\prod_p p^{a_p}\;\;$   and   $\;b=\prod_p p^{b_p}\;\;$   their least common multiple and greatest common divisor are given by the formulas and Since this gives In fact, any 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. Using the same examples as above: The positive integers may be partially ordered by divisibility: if a divides b (i.e. if b is an integer multiple of a) write ab (or equivalently, ba). (Forget the usual magnitude-based definition of ≤ in this section - it isn't used.) 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: The following pairs of dual formulas are special cases of general lattice-theoretic identities. It can also be shown that this lattice is distributive, i.e. that lcm distributes over gcd and, dually, that gcd distributes over lcm: This identity is self-dual: Let D be the product of ω(D) distinct prime numbers (i.e. D is squarefree). Then where the absolute bars || denote the cardinality of a set. 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 (i.e. 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). In principal ideal domains, one can even talk about the least common multiple of arbitrary collections of elements: it is a generator of the intersection of the ideals generated by the elements of the collection.