134 



THE POPULAK EDUCATOR. 



LESSONS IN ARITHMETIC. IX. 



LEAST COMMON MULTIPLE. 



1. ONE number is called a multiple of another when it can be 

 divided by the latter without a remainder. 



Tims, a measure and a multiple are the converse of each other. 

 K a number divides another without remainder, it is said to be 

 a measure of it, and the latter number is said to be a multiple 

 of the first. 



A common multiple of two or more numbers is a number 

 which can be divided by each of them without a remainder. 

 It will clearly be a composite number, of which each of the g'iven 

 numbers must be a factor, for it could not otherwise be divided 

 .by them. 



The same numbers may clearly have an infinite number of 

 common multiples, for any one common multiple having been 

 found, another may be obtained by multiplying it by any 

 number. 



The continued product of two or more numbers will always 

 .give a common multiple of those numbers. 



The least common multiple of two or more numbers is tha 

 least number which can be divided by each of them without a 

 remainder. 



Thus. 70 is the least common multiple of 2, 5, and 35. 



2. The least common multiple of two or more numbers is 

 evidently composed of the continued product of all the different 

 prime factors which compose the given numbers, each one being 

 repeated as often as the greatest number of times it occurs in 

 <my one of the numbers. For if it did not contain all the prime 

 factors of any one of the numbers, it could not bo divided by 

 that number. 



On the other hand, if any prime factor were employed more 

 times than it is repeated in any one of the given numbers, it 

 would not be the least common multiple. 



For the sake of brevity the words " least common multiple " 

 are sometimes written L. C. M. 



3. EXAMPLE. Find the L. C. M. of 12, 126, and 735. 

 These are respectively equal to 



2x2x3, 2x3x3x7, 3x5x7x7. 

 Now 2. 3, 5, 7 are all the different prime factors which occur 

 in any of the numbers ; and the greatest number of times which 

 .2 occurs is twice namely, in the first ; the greatest number 

 which 3 occurs is twice namely, in the second ; 5 only occurs 

 once namely, in the third ; and the greatest number of times 

 which 7 occurs is twice namely, in the third. Hence the 

 ,L. C. M. required will be 



2x2x3x3x5x7x7; that is, 8820. 



4. The process, then, of finding the least common multiple of 

 two. or more numbers is reduced to that of splitting up the 

 numbers into their prime factors. 



This may be effected, however, by a more convenient method 

 of arrangement than splitting each number separately into 

 factors would be. for which we give the following 



Rule for finding the least common multiple of two or more 

 numbers. 



Write down tha numbers in a straight line apart from each 

 other. Divide by the least number which is a measure of two 

 or more of them, and set down the quotients and the undivided 

 numbers in a line below. Take again the least number which 

 is a measure of two or more of these numbers last set down, and 

 perform the same operation as before. Continue it until there 

 are no two numbers which can be divided by any number greater 

 than unity. The continued product of all the divisors, and the 

 numbers set down in the last line, will be the least common 

 multiple required. 



5. EXAMPLE. To find the L. C. M. of 12, 42, 72, and 84. 

 The process will be sufficiently understood from the following 



Working : 



2 ) 12, 42, 72, 84 



2) 6,21, 36,42 



3 ) 3, 21, 18, 21 

 7) 1, 7, 6, 7 



1, 1, 6, 1 

 Hence the L. C. M. is 6 X 7 X 3 X 2 X 2 ; that is 504. 



This method of arrangement evidently gives the greatest 

 number of times which each prime factor occurs in any one of 

 the given numbers. Thus 2 occurs three times in 72, 3 occurs 

 twice in 72, and 7 occurs only once viz., in 42 and 84. 



EXERCISE 21. 

 Find the least common multiple of the following numbers :- - 



15 and 45. 

 63 and 18. 

 6, 9, and 15. 



8, 16, 18, and 24. 



9. 15. 12, 6, and 5. 

 5, 10, 8, 18, and 15. 

 24, 16, 18, and 20. 

 36, 25, 60, 72, and 35. 

 27, 54, 81, 14, and 63. 

 72, 120, 180, 24, and 36. 

 375, 850, 3400, and 5085. 



7, 11, 13, and 5. 



13. 1, 2, 3, 4, 5, 6, 7, 8, and 9. 



14. 657, 350, 876, 1095. 2190, 



and 5795. 



15. 42, 12, 84, and 72. 



16. 9, 12, 72, 36, and 144. 

 J7. 8, 12, 20, 24, and 25. 



18. 63, 12, 84, and 7. 



19. 54, 81, 63, and 14. 



20. 75, 120, and 300. 



21. 96, 144, and 720. 



22. 256, 512, and 1728. 



23. 375, 850, and 3400. 



LESSONS IN GERMAN. VIII. 



SECTION XVI. USE OP THE DEFINITE ARTICLE; 

 PROPER NAMES, ETC ETC. 



THE plural of 2Uann is SJJanntr; except in compounds, where it 

 is generally Scute ( XV. Note), as Santtmtm. countryman ; Sank- 

 (eutc, country-people. 3intmevmann. carpenter: Bimmcrlcute, car- 

 penters. J&iiuptmann, captain ; auptlcutc, captains, tfaufmann, 

 merchant; .ftauflcute, merchants. 



SSoIf corresponds mainly to our word people. Unlike this, 

 however, it has different forms for the two numbers, as : Sir 

 gmnjofen fink cin lebtyafteS 93 o ( f ; the French arc a lively people. 

 35tt gurften fd?h>ergen, unk tag 93 olf fetket ; the princes revel, and the 

 people suffer. 2ll(e 93 5 If er auf (Srken, 1 SDJcfcS xviii. 18 ; all the 

 nations of the earth, Genesis xviii. 1 8. 



The word one, as a pronoun, is, in English, often inserted 

 after an adjective, to avoid the repetition of the noun ; in Ger- 

 man, however, the adjective in such a case stands alone, as : 

 @r l)at cinm gutcn ut. unk tcb fiabc eincn fcftjcdbtcn ; he has a good hat, 

 and I have a bad (one). 3cty fjafre gutc itte, unb er ^at fcb, (ed)te ; I 

 have good hats, and ho has bad (ones). 6r Ijat gutcn SBctn, unb tcty 

 [;abc fd;lccf)ten ; he has good wine, and I have bad (wine). 



The adjective and participle preceded by an article are often 

 used substantively, as well in the singular as in the plural, as : 

 cr Sufnekcne (Sect. IX. 2) tft gtiicflid) ; the contented (man) is 

 happy. 2)te Sufrtekene tft glucfltd; ; the contented (woman.) is 

 happy. S)te Sufrtekenen ftnb gluctlicfy ; the contented are happy. 

 (Sin 3ufrickcnet (Sect. X.) tft gliicf licty ; a contented (man) is happy. 

 er tcvbcnke, kit tcrbenke; the dying (man), the dying (woman). 

 2)te cbentcn ; the living. 



1. Adjectives in German are often, by means of the definite 

 article, converted into abstract nouns, as : (5r scrd)rt ka g^cne ; 

 he adores the beautiful. 



2. The use of the definite article before nouns, taken in a 

 general sense, is much more frequent than in English, as : f>er 

 iget 'ft flinf : the tiger is agile. 2)cr Clamant ift em 6fcc(ftcin ; the 

 diamond is a precious stone. a QMt tft cin efcelS (DJctatt ; (the) 

 gold is a precious metal. >tc Suft tft cin temcnt ; the air is an 

 element. a SSaffer ift etn (Element ; (the) water is an element. 

 <U eete if} unftcrbltcf> ; the soul is immortal. S5er SHenfty tft 

 fterblicfi; (the) man is mortal. Stte gaulfydt tft cin Saftcr; (the 1 ) idle- 

 ness is a vice. 



The plural is used in the same manner, as: 3Me Sigcr fink flinf; 

 (the) tigers are agile. 



3. The definite article is sometimes used instead of the pos- 

 sessive pronouns, as : @r fyat etn 33ucf> in k e r ank ; he has a 

 book in the (his) hand. !Dn8 JUnk tft bet k e m SSatcr ; the child is 

 with the (its) father. 



4. Proper names and titles are often preceded by the defi- 

 nite article, as : S8o ift ker chmcJ> V where is (the) Henry ? er 

 tf atfer ->cinrtcf) , the Emperor Henry. 2>er tfcnig dnnc$ ; (the) King 

 Henry. 



The definite article likewise commonly precedes the adjective 

 qualifying a proper name, as : 3)tc fd;i?nc ^clcne ; the beautiful 

 Helen. 5?er arme JJJicfjark ; (the) poor Eichard. 



The article is also generally used before the word 4ule, 

 flirctye, 3)larft, 2)iub,k, :c., as : r tft in fccr cfjule; he is (in) at (the) 

 school, r tft in bcr tfircfce ; he is (in) at (the) church. @r ift auf 



