between it and 24, and write the result beneath. The numbers in the lower line having no factor in common, we carry the process no further. The continued product of the number cut off by the numbers in the lower line gives 1056, the least common multiple, as by the other methods. In this instance we cut off the 24, but either the 32 could have been separated from the rest, or the 44 cut off, and the needless factors striken out with like result. If, however, we had cut off the 44, the numbers placed in the second line would have contained factors common to each other, so that it would have been necessary in that line to have cut off and stricken out factors as before. The reason for this abridged process is, that by the separating off, and by the striking out of factors, we get rid, in an expeditious way, of the factors not required to form the least common multiple sought. Rule 1. — Resolve the given numbers into their prime factors. The product of these factors, hiking each factor only the greatest number of times it occurs in any of the numbers, will be the least common multiple. Or, Rule 2. — Having arranged the numbers on a horizontal line, cancel such of them as are factors of any o f the others, and separate some convenient one from the rest. Reject from each of the numbers remaining the greatest factor common to it and that number, and write the result in a line below. Should there be in the second line numbers having factors in common, proceed as before; and so continue until the numbers written below are prime to each other. The continued product of the number or numbers separated from the others with those in the last line will be the least common multiple. Note 1. — Some give a preference to the following rule for finding the least common multiple: Hiving arranged the numbers on a horizontal line, divide by such a prime number as will exactly divide two or more of them, and write the quotients and undivided numbers in a line beneath. So continue to divide until the quotients shall be prime to each other. Then the product of the divisors and the numbers of the last line will be the least common multiple. Note 2. — The least common multiple of two or more numbers that are prime to each other is found by multiplying them together (Art. 202). Note 3. — When a single number alone is prime to all the rest, it may be separated off, and used only as a factor of the least common multiple sought. Note 4. — When the least common multiple of several numbers, and all the numbers except one, which is prime to the others, are given, to find the unknown number, divide the least common multiple given by that of the known numbers (Art. 202). Examples. 2. What is the least common multiple of 3, 13, 37, and 91. 3. "What is the least common multiple of 9, 14, 30, 35, and 47? Ans. 29610. OPERATION. 9 1 4 3 0 ( 3 5 4 7. 9 <| »" 47 X 35 X 6 X 3 = 29010 Ans. 4. What is the least common multiple of 6, 8, 10, 18, 20, and 24? 5. What is the least common multiple of 14, 19, 38, and 57? Ans. 798. 6. What is the least common multiple of 20, 36, 48, and 50? Ans. 3600. 7. What is the least common multiple of 15, 25, 35, 45, and 100? Ans. 6300. 8. What is the least common multiple of 100, 200, 300, 400, and 575? Ans. 27600. 9. The least common multiple of 1, 2, 3, 4, 5, 6, 8, 9, and one other number prime to them, is 2520. What is that other number? Ans. 7. 10. What is the least common multiple of 18, 24, 36, 126, 20, and 48? • 11. I have four different measures; the first contains 4 quarts, the second 6 quarts, the third 10 quarts, and the fourth 12 quarts. How large is a vessel, that may be filled by each one of these, taken a certain number of times full? Ans. 60 quarts. 12. What is the smallest sum of money with which I can purchase a number of oxen at $ 50 each, cows at $ 40 each, or horses at $ 75 each? Ans. $ 600. MISCELLANEOUS EXAMPLES. 1. How many times does 7 occur as a factor of 6174? Ans. 3 times. 2. Required the largest prime factor of 5775. 3. Required the largest composite factor of 19929. Ans. 6643. 4. Required the quotients of 2338 divided by its two prime factors next larger than 1. Ans. 1169; 334. 5. Required all the prime numbers that will divide 17385 without a remainder. 6.° A farmer has 3000 bushels of grain; which are the three smallest-sized bags, and the three largest-sized bins, holding an exact number of bushels, that will each measure the same without a remainder? Ans. Bags of 1, 2, or 3 bushels each; and bins of 1500, 1000, or 750 bushels each. 7. A teacher having a school consisting of 152 ladies and 136 gentlemen, divided it in such a manner that each class of ladies equalled each class of gentlemen, and the classes were the largest the school would admit of, and have them all of the same size. Required the number of classes, and the number in each class. Ans. 19 classes of ladies, 17 classes of gentlenien, and 8 pupils in a class. 8. At noon the second, minute, and hour hand of a clock are together; how long after will they be again, for the first time, in the same position? 9. J. Porter has a four-sided garden, the first side of which is 348 feet in length; the second, 372 feet; the third, 444 feet; and the fourth, 492 feet. Required the length of the longest rails that can be used in fencing it, allowing the end of each rail to lap by the other 9 inches, and all the panels to be of equal length; also, the number of rails, if 5 rails be allowed to each panel. Ans. Length 12ft. 9in.; and 690 rails. 10. L. Ford has 5 pieces of land, the first containing 3A. 2R. lp.; the second, 5A. 3R. 15p.; the third, 8A. 29p.; the fourth, 12A. 3R. 17p.; and the fifth, 15A. 31p. Required the largest sized house-lots, containing each an exact number of square rods, into which the whole can be divided. Ans. 1A. 27p. each. 11° What three numbers between 30 and 140 have 12 for their greatest common divisor, and 2772 for their least common multiple. Ans. 36, 84, and 132. 12. Four men, A, B, C, and D, are engaged in making regular excursions into the country, between which each stays at home just 1 day; and A is always absent exactly 3 days, B 5 days, and C and D 7 days. Provided they all start off on the same day, how many days must elapse before they can all be at home again on the same day? , Ans. 23 days. COMMON FRACTIONS. 204. A Fraction is an expression denoting one or more equal parts of a unit. 205i A fractional unit is one of the equal parts into which the whole thing or integral unit has been divided. Thus halves, thirds, &c, being equal parts of integral units or whole things, are fractional units. 206. The unit of a fraction is the unit or whole thing from which its fractional parts have been derived. 207. A Common Fraction is expressed by two numbers one above the other, with a line between them. 2C8. The number below the line is called the denominator. It shows into how many parts the whole number has been divided. It gives name to the fraction and value to the fractional unit. Thus, in the expression f, the denominator is 7, indicating that the unit of the fraction has been divided into 7 equal parts, and that the value of the fractional unit is one seventh. The number above the line is called the numerator. It shows how many parts have been taken, or numbers the fractional units expressed by the fraction. Thus, in the expression f, the numerator is 2, indicating that the fractional unit, which is one seventh, has been taken 2 times. 209. The terms of a fraction are its numerator and denominator. Thus, the terms of the fraction § are the numerator 2 and the denominator 3. 210. A proper fraction is one whose numerator is less than the denominator; as $, j. 211 i An improper fraction is one whose numerator is equal to, or greater than, the denominator; as \, 212. A mixed number is a whole number with a fraction; as 3i, Ui, 901. 213. A simple or single fraction has but one numerator and one denominator. It may be either proper or improper; as 214. A compound fraction is a fraction of a fraction, or two or more fractions connected by the word of; as J of ^ of -jv, I off of $. 215. A complex fraction is a fraction having a fraction or a mixed number for its numerator or denominator, or both; as i ]_ 8* H ?' H' 13' 4i 216i A fraction is an expression of division; the numerator answering to the dividend, and the denominator to the divisor, (Art. 67); and the value of a fraction is the quotient arising from the division of the numerator by the denominator (Art. 80). Thus, in the fraction the numerator 15 is the dividend, the denominator 7 is the divisor, and the value expressed 2), or the quotient arising from the division of the 15 by the 7. 217. Since a fraction is an expression of division, it follows, 1. That, if the numerator be multiplied, or the denominator be divided, by any number, the fraction is multiplied by the same number (Art. 81). 2. That, if the numerator be divided, or the denominator multiplied, by any number, the fraction is divided by the same number (Art. 82). 3. That, if the numerator and denominator be both multiplied, or both divided, by the same number, the fraction will not be changed in value (Art. 83). REDUCTION OF COMMON FRACTIONS. 218. Reduction of fractions is the process of changing their form of expression without altering their value. 219. A fraction is in its lowest terms, when its numerator and denominator are prime to each other (Art. 166). 220. To reduce a fraction to its lowest terms. Ex. 1. Reduce J| to its lowest terms. Ans. First Operation. By dividing both tenris of the fraction by 4 ) jf = -tv 4, a factor common to them both, it is re 4 ) T% = £ Ans. duced to ^. Dividing both terms of 75 by 4, a factor common to them both, it is re |