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.
The RBDe 560 (in the old naming style, the RBDe 4/4) and its derivatives provide motive power for S-Bahn, suburban, and regional traffic on the Swiss Federal Railways (SBB) network. The derivative versions belong to the SBB as well as various private railroads. The locomotive and its matching Bt model Steuerwagen (translation: control car/cab car/driving trailer) form compositions generally known as the Neuer Pendelzug (New Commuter Train), which is the source of the acronym NPZ. An NPZ trainset usually includes one or more intermediate cars.
In 1984 four pre-series sets (each consisting of a motor car and a driving trailer) were delivered. Ordered in 1981, they originally bore the RBDe 4/4 designation and road numbers 2100-2103. All four trainsets (RBDe 560 + Bt) were delivered in different color schemes, one of which was the livery used for the main series (blue over white sides, yellow doors, and red faces). The striking contrast to the green color scheme of previous SBB passenger stock led to the name Kolibri (Hummingbird), which is, however, rarely used. Nearly all the RBDe 560 sets have been named after smaller municipalities along the lines served by these trainset received the appropriate coat of arms.
A full order for 80 trainsets followed. A few years later an additional order for a further 42 trainsets was placed. 6 trainsets were ordered by private railroads (Südostbahn (SOB), PBr, MThB, Montafonerbahn, etc.), resulting in a total production of 134 series trainsets. The last were put in service in 1996.
For cost reasons, up to the procurement of new low floor intermediate cars Domino starting in 2006, modernized Einheitswagen (EW) I and II coaches are used to achieve the desired passenger capacity. A standard set has a second class and a combined first/second class car, but sets with zero or up to three intermediate cars are being operated.
The motor cars have a power output of 1650 kW, a maximum speed of 140 km/h, a weight of 70 tons, and are 25 meters long. The series (560 000-083 through 560 100-141) forms the largest group of regional/suburban traffic motor cars of SBB, with 126 vehicles.
In May 2006 a modernization program was begun by the SBB for 120 of its trainsets. The motor cars and driving trailes are being refurbished and then matched with new air-conditioned low floor intermediate coaches to form Domino trains. The existing EW I and II coaches will, after well over 40 years of service, be scrapped. Further use of the trucks on these coaches is being considered. The 188 new intermediate cars are to be delivered coinciding with the modernization, which is forecast to take 7 years and should be complete by 2013.
The NPZ color scheme mentioned above is generally common today. The second production series (560 100 - 560 141) has the blue window strip extended all the way to the driver's compartment. In addition, numerous examples have logos of the various Swiss S-Bahn systems (Zürich, Basel, Bern, Ostwind, Réseau Express Vaudois, etc.) applied to the side of the locomotive near the driver's compartment.
Those units operating the Ticino-Lombardy Treni Regionali Ticino Lombardia (TILO) service have the full-size SBB logo removed, which is replaced by the TILO logo above the windows. The logos of SBB and Trenitalia (joint owners of TILO) are applied in a smaller format.
The two motor coaches assigned to the Glarner Sprinter service (560 120 and 560 123) have that logo applied alongside the SBB logo, as well as a special paint scheme consisting of various highlights to the base NPZ livery.
One set, RBDe 560 131 with Bt 29-35 931 got a special livery "train des vignes" for the Vevey–Puidoux-Chexbres line plus a stop on request system. RBDe 560 132/Bt 932 were fitted equally as replacement set but painted in the normal livery.
13 trainsets are assigned to the RegionAlps service.
In 2003 six RBDe 560 (105, 127-128, 133-135) were converted to RBDe 561 (000-005) so that they could operate on the Wiesental section of the Basel S-Bahn (Basel - Zell im Wiesental, Germany). With the subsequent delivery of 10 RABe 521 (Stadler FLIRT) specifically for this network, these six RBDe 561 were returned in May 2006 to the Swiss side of the Basel S-Bahn and now operate again as RBDe 560. However, these trainsets remain recognizable because they still have certain German radio equipment installed.
At the beginning of 1997, only one year after delivery, six more RBDe 560 (136-141) were converted to operate on the French section of the Basel S-Bahn (between Basel and Mulhouse). These bi-current trainsets received the designation RBDe 562 (000-005), and are capable of operating on the 15 kV/16.7 Hz SBB standard as well as the 25 kV/50 Hz SNCF standard.
The Südostbahn (SOB) also ordered four two car NPZ sets (RBDe 566 + ABt), which were delivered out of the first production run in 1995 receiving fleet numbers RBDe 566 400-403. The SOB opted not to convert any intermediate cars for use with these trainsets. To provide first class accommodation the SOB control trailers have a first class compartment, thus the designation ABt, as opposed to SBB's second class only Bt. The motor cars are, except for the livery, identical to those of the SBB. The SOB has since converted several intermediate cars and operates the trainsets are operated with them.
After the merger of the original SOB with the Bodensee-Toggenburg-Bahn (BT) to the new SOB, these four trainsets were repainted in the new livery of the SOB in 2003. They now operate throughout the SOB network as RBDe 566 077-080.
For services on the Seehas line (the area near Konstanz, Germany), the Mittelthurgaubahn (MThB) in 1993 took four NPZ trainsets off the SBB's production line. These 4 trainsets came from the first production series, and were delivered to the MThB in 1994 numbered as RBDe 566 631-634. These 4 trainsets would have been the ninth to twelfth for the SBB, which subsequently increased its total order by four.
These four MThB trainsets were also delivered with one new intermediate car each. Two additional driving trailers were used with older ABDe 536. Because they operate into Germany as well as within Switzerland, the vehicles were equipped with the Indusi train safety system and a wider pantograph shoe.
Upon the dissolution of the MThB, the trainsets went to the successor company Thurbo and were operated on the Seehas German section by Thurbo's German subsidiary EuroThurbo and later SBB GmbH. In 2006, nine Stadler FLIRT (RABe 526 651-659) took over the Seehas service. Thurbo subsequently sold the four trainsets plus the two extra driving trailers to SBB. They work now as RBDe 561 171-174/AB 30-35 671–674/Bt 29-35 971–976 in the S-Bahn Luzern system, partly over BLS tracks.
The Montafonerbahn, which is headquartered in Schruns, Austria, operates two two-piece NPZ sets. The first was built in 1990, following the SBB series 2100-2183, and the second in 1993, before the SBB second series.
The Chemin de fer Pont Brassus (PBr) has never possessed its own locomotives, but has always subcontracted operation to the SBB, which guarantees through service between Vallorbe–Le Pont (SBB) and the PBr route Le Pont-Le Brassus in the Vallée de Joux.
The "+2" in the first series production count are the two trainsets which were sourced in 1989 the line and went to the PBr. The locomotives originally received the designation RBDe 4/4 2184-2185. Only small labels referring to the PBr as the owner of these vehicles distinguished them, while the SBB remains responsible for their maintenance and operation. After the January 2001 merger of the PBr with the Chemin de fer Yverdon - Ste Croix (YSteC), which formed TRAVYS, the trainsets were repainted into that company's new livery and the motor coaches renumbered RBDe 568 384-385.
There are several Swiss trainsets which are very similar to the RBDe 560 series, but are not identical. These were jointly developed by BLS and Bodensee-Toggenburg-Bahn (now Südostbahn) and first delivered in 1982. The SBB design was derived from these trains. They operate on various private railways as BLS RBDe 565, RBDe 566 II, Südostbahn RBDe 566 and TRN/tpf RBDe 567.
This article was partially translated from the German language version of July 2006. It has been amended by the authors of the German article.
Comparable transactions is one of the conventional methods to value a company for sale. The main approach of the method is to look at similar or comparable transactions where the acquisition target has a similar business model and similar client base to the company being evaluated. This approach is fundamentally different from that of DCF valuation method, which calculates intrinsic value.
For instance, Providence Equity Partners acquired Virtual Radiologic Corporation, which is an online clinic that provides radiologist analysis through a virtual network. It was sold for a price of $282 million and an enterprise Value of $242 million. To evaluate a similar unsold company, we would look at what are called the transaction multiples. One popular transaction multiple is EV/EBITDA. For Virtual Radiologic Corporation, the EBITDA at the time of the transaction was $20 million, giving an EBITDA multiple of 12.1x. A similar unsold company, which has EBITDA at $10 Million could expect to be sold for $120 million. In some market segments, the companies do not have high EBITDA, and sometimes a multiple based on revenues (EV/sales) is used instead. To get a more accurate valuation, one should look at the multiples of more than one similar deals that are relatively recent since multiples do change from year to year.
In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n , which is called the multiplier or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer with no remainder. If a and b are both integers, and b is a multiple of a, then a is called a divisor of b.
14, 49, -21 and 0 are multiples of 7, whereas 3 and -6 are not. This is because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and -21, while there are no such integers for 3 and -6. Each of the products listed below, and in particular, the products for 3 and -6, is the only way that the relevant number can be written as a product of 7 and another real number:
Unlike in many European parliamentary cabinets, in the Cabinet of the United States, it is generally less common for a cabinet secretary to hold multiple cabinet-level positions over the years. This is a list of the exceptions:
The following served in multiple cabinet positions in the Confederate States:
The following have served in cabinet positions in both the United States and Confederate States:
In arithmetic and number theory, the least common multiple
(also called the lowest common multiple
or smallest common multiple
) of two integers a
, usually denoted by LCM(a, b)
, is the smallest positive integer that is divisible by both a
. If either a
is 0, LCM(a
) 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
as lcm( a
). Some older textbooks use [ a
What is the LCM of 4 and 6?
Multiples of 4 are:
and the multiples of 6 are:
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
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
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,
) is a divisor of both a
, it's more efficient to compute the LCM by dividing before
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
computation). Because gcd(a
) is a divisor of both a
, 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.
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
, ..., xn
> 1. The algorithm proceeds in steps as follows: on each step m
it examines and updates the sequence X
( = (x1
(, ..., xn
(1) = X
, where X
( is the m
th 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
(. Assuming xk0
( is the selected element, the sequence X
( is defined as
In other words, the least element is increased by the corresponding x
whereas the rest of the elements pass from X
( to X
The algorithm stops when all elements in sequence X
( 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
, ... are non-negative integers; for example, 84 = 22 31 50 71 110 130 ...
Given two integers
their least common multiple and greatest common divisor are given by the formulas
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
(i.e. if b
is an integer multiple of a
) write a
(or equivalently, b
). (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:
be the product of ω(D
) distinct prime numbers (i.e. D
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
be elements of a commutative ring R
. A common multiple of a
is an element m
such that both a
(i.e. there exist elements x
such that ax
). A least common multiple of a
is a common multiple that is minimal in the sense that for any other common multiple 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
can be characterised as a generator of the intersection of the ideals generated by a
(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.