126 



THE POPULAR EDUCATOE. 



so found, which evidently will be the middle of c x, is the centre 

 of the three. Now join T with D, and divide T r> into ten parts 

 (the sum of 1, 2, 3, and 4), and take four next Y and six next 

 D. This last point, z, is the centre of all the given forces. Try 

 your own hands now on the following Examples, and in the next 

 lesson we shall have for subject the centre of gravity, which is 

 a centre of parallel forces. 



Examples. 



1. Three equal parallel forces act at the comers of a triangle ; find 

 the centre through which their resultant passes. 



2. A force of a pound is applied to one end of a beam, of three at 

 the other, and of two at the middle j find tlie centre of these forces, 

 they being parallel to each other. 



3. A weight of one pound and three-quarters hangs from one end of 

 a rod which is two feet in length, and of three and a-half from the 

 other ; find the magnitude of the resultant, and the centre of parallel 

 forces. 



4. A door is seven feet high and three feet wide, and the centres of 

 its hinges are distant one foot from its ends. A force of twenty-three 

 pounds is applied along its upper edge, pulling it off its hinges, and 

 one of thirty-seven along the lower. Find the strains on the hinges. 



LESSONS IN ARITHMETIC. VIII. 



GREATEST COMMON MEASURE. 



1. A composite number, as already defined (see Lesson VI., 

 Art. 2), is one which is produced by multiplying two or more 

 numbers or factors together. 



A prime number is one which cannot be produced by multi- 

 plying two or more numbers together ; it cannot, therefore, bo 

 exactly divided by any ivlwle number except unity and itself. 

 Thus 1, 2, 3, 5, 17, 31, etc., are prime numbers, or primes, as 

 they are sometimes called. 



A measure of any given number is a number which will divide 

 the given number exactly without a remainder. Thus, 3 is a 

 measure of 9, 25 is a measure of 75. 



A common measure of two or more numbers is a number which 

 will divide each of them without a remainder. Thus, 2 is a 

 common measure of 6, 8, 12, 18, 30, etc. 



The greatest common measure of two or more numbers is 

 the greatest number which will divide them all without a 

 remainder. Thus, 9 is the greatest common measure (or, as it 

 is sometimes written for shortness, the G. C. M.) of 18, 27, 36, 

 and 45. 



2. To find tlie greatest common measure of iivo given numbers. 



EULE. Divide the greater by the less, then the preced- 

 ing divisor by the remainder, and so on, until there is no 

 remainder. The last divisor will be the greatest common mea- 

 sure required. 



EXAMPLE. To find the greatest common measure of 532 and 

 1274. Arrange the process thus : 



532)1274(2 

 1064 



210)532(12 

 ' 420 



112 ) 210 ( 1 

 112 



93)112(1 



14)98(7 



Here, in accordance with the rule, we divide 1274 by 532, 

 wldch gives a remainder 210; then 532 (the preceding divisor) 

 by 210, giving a remainder 112 ; again 210 (the preceding 

 divisor) by 112, which gives a remainder 98 ; then 112 (the pre- 

 ceding divisor) by 98, which leaves a remainder 14 ; and lastly, 

 98 by 14, which gives no remainder. 14, therefore, according 

 to the rule, is the greatest common measure of 532 and 1274. 



3. To find, the greatest common measure of three or more given 

 numbers. 



RULE. Find the greatest common measure of two of them ;. 

 then find that of the common measure thus obtained and of the 

 third ; then that of this common measure and the fourth, and so 

 on. The last obtained will be the greatest common measure of 

 the given numbers. 



EXAMPLE. Find the greatest common measure of 204, 357, 

 and 935. 



First, we find the greatest common 204 ) 357 ( 1 

 measure of 204 and 357 to be 51, by the 204 



rule given for two numbers. 



153)204(1 

 153 



51)153(3 

 153 



51 ) 935 ( 18 

 51 



425 

 408 



17)51(3 

 51 



Next, we find the greatest common measure of 51 

 and 935, which we see to be 17. 



Hence, according to the rule, 17 is the greatest 

 common measure of 204, 357, and 935. 



We do not give the reasons for the truth of the 

 foregoing rules, as they cannot be satisfactorily 

 established without the aid of algebra. 



4. The above rules are infallible methods for finding the 

 greatest common measure of two or more numbers. In practice, 

 however, we can frequently dispense with these operations, and 

 determine the greatest common measure by inspection, or by 

 splitting up the numbers into their elementary or prime 

 factors. 



It is evident that if two or more numbers have a common 

 measure at all, they must be composite numbers, i.e., capable of 

 being separated into factors. If any given numbers be sepa- 

 rated into prime factors, the greatest common measure will 

 evidently be the product of all the factors which are common to 

 each of the given numbers. 



Thus, 75, 135, and 300, when separated into their prim 

 factors, are respectively 



3x5x5, 3x5x9, and 2x2x3x5x5 



Now, the factors which are common to all of these are 3 and 5, 

 and therefore 3 X 5 that is, 15 is the greatest common; 

 measure of 75, 135, and 300. 



5. We subjoin a 



Rule for dividing a composite number into its prime factors. 



Divide the given number by the smaller number, which will 

 divide it without a remainder ; then divide the quotient in the 

 same way, and continue the operation until the quotient ia 

 unity. The divisors will be the prime factors of the given 

 number. 



The reason of the truth of the above rule may be thus ex- 

 plained : 



Every division of a number, where there is no remainder, 

 resolves it into two factors namely, the divisor and quotient. 

 But in the above rule the divisors in each case are the smallest 

 numbers which will divide the given number and the successive 

 quotients without a remainder : hence they are all prime num- 

 bers, and the division is continued until the quotient is unity. 

 Hence, clearly, the product of all these divisors (which are all 

 primes) will be equal to the original number. In other words, 

 these divisors are the prime factors of the given composite 

 number. 



EXAMPLE. Resolve 16170 into its prime factors. Arrange 

 the process thus \ 



2)16170 



