GREATEST COMMON DIVISOR 



(d) Therefore, in seeking the G. C. D. we 

 seek for the highest common factor. 



(1) Find the G. C. D. of 16, 40, 72. Fac- 

 toring, we see: 



16 = 2X8 

 40 = 5X8 

 72 = 9X8 



and recognize 8 as the G. C. D. 



(2) Find the G. C. D. of 75, 45, 105, 135. 



75 = 5X5X3 



45 = 3X5X3 

 105 = 7X5X3 

 135 = 3X5X3X3 



Here we find 5X3, or 15, to be the G. C. D. 



(3) Find the G. C. D. of 672, 504, 924. 



672=(2X2)X2X2X2X3X 7 

 504=(2X2)X2X 3X3X7 



924=(2X2)X 3X 7X11 



We see that (2X2) is common, and 3 is com- 

 mon, and 7 is common to all the numbers as a 

 divisor; therefore (2X2X3X7), or 84, is the 

 G. C. D. 



If the young student has difficulty in seeing 

 this, he will find it helpful to write the factors 

 in this way: 



672 = 2x 2X2X(2X2X3X7) 

 672= 8 X 84 

 504 = 2x 3 XC2X2X3X7) 

 504= 6 X 84 

 924= 11 X(2X2X3X7) 

 924= 11 x 84 



Below it is seen that we may break up the 

 numbers into their composite factors and from 

 these find the G. C. D. 



225 = 15X15 

 150 = 10X15 

 375 = 25X15 



15 is common, and there is the common fac- 

 tor 5 in 15, 10 and 25. So we know that 75 

 is the G. C. D. and we see: 



225 = 3X75 

 150 = 2X75 

 375 = 5X75 



We may find the common prime factors in 

 this way: 



51225 150 375 



2595 GREAT PALLS 



Find the G. C. D. of 4633, 697 and 943. 



5 [45 

 3 |9 



30 



75 



IB 



325 



5, 5 and 3 are the common divisors, and their 

 product, 5X5X3, or 75, is the G. C. D. 



When numbers are prime to each other, their 

 G. C. D. is 1; in other words, they have no 

 G. C. D.; for example 7, 20 and 9. 



When the factors of several numbers cannot 

 be found at sight, nor by using tests for divisi- 

 bility of numbers, the following method is used 

 for finding the G. C. D.: 



(a) 69714^33 



4182 1 



451 1 



246|45l 



246 1 



^05124$ 



205 



(b) 



23 

 9~43 

 82 

 123 

 123 



41 is the G. C. D. of 4633, 697 and 943. 



GREAT FALLS, MONT., noted for its great 

 smelting works, is situated a little northwest 

 of the geographical center of the state, in 

 Cascade County, of which it is the county seat, 

 and on the Missouri River. Helena is ninety- 

 nine miles southwest; Butte, also southwest, is 

 173 miles distant; and Billings is 235 miles 

 southeast. The Great Northern Railway, built 

 to the city in 1887, and the Chicago, Mil- 

 waukee & Saint Paul, constructed to this point 

 in 1913, provide transportation facilities. The 

 city was settled in 1883, was incorporated in 

 1887 and named for the great falls of the 

 Missouri River, which at this point have a 

 total descent of 500 feet. Its population, 

 chiefly American, was 13,948 in 1910, giving 

 the city second rank in the state (after Butte). 



Great Falls is an attractive city, with wide, 

 shaded streets and a park system covering 640 

 acres. It is located in a territory famous for 

 scenic beauty, which is productive as. well as 

 picturesque. Gold, silver, copper, lead, iron, 

 bituminous coal and sandstone are found in 

 abundance; upon these resources depend the 

 leading industries of the city. The smelting 

 works of one mining company alone employ 

 1,250 people, the monthly pay roll amounting 

 to between $125,000 and $150,000. A remark- 

 able structure in connection with the smelting 

 industry is a smokestack, claimed to be the 

 largest in the world. It stands on a hill, and 

 rises 506 feet above the foundation, its height 

 being exceeded only by two buildings in New 

 York City, the Washington Monument and 

 the Eiffel Tower. The immense dam at Blark 

 Eagle Falls furnishes abundant water power for 

 manufacturing purposes, and the number of 

 flour and oatmeal mills, beet-sugar and can- 

 ning factories suggests the agricultural wealth 

 of the vicinity; the yield of alfalfa is most 



