Find the L. C. M. of 14, 18, 21, 27, 28, 126. 14= 2 × 7, 18 = 2 × 32, 21 = 3 × 7, 27 = 33, 28=22 × 7, 1262 x 32 × 7. = Hence, the L. C. M. 22 x 38 x 7 = 756. 241. The L. C. M. of 14, 18, 21, 27, 28, 126, may be found as follows: Hence, the L. C. M. 22 x 33 x 7, or 756. = Since 14 is contained in 28, it is omitted, for any multiple of 28 is also a multiple of 14. Likewise 18, which is contained in 126, and 21, which is also contained in 126, are omitted. The even numbers are divided by 2; the quotients and the odd numbers are written below the horizontal line. The operation of dividing by 2 is repeated; the quotients and the odd numbers are written below. Of the resulting numbers (27, 7, 63) 7, which is contained in 63, is omitted, and division by 3 gives the quotients 9 and 21. The second division by 3 gives the quotients 3 and 7, which are seen to be prime to each other. By this process the prime factors of each of the given numbers are obtained as divisors or as last quotients; therefore, the product of the divisors and last quotients is the L. C. M. required. 242. If the given numbers are large and contain no prime factors that can readily be detected, it is best to obtain the common factors by the process for finding the G. C. M. under like circumstances. Find the L. C. M. of 1247 and 1769. 1247)1769 (1 2 522 9 261 29)1247(43 116 87 87 Hence, the G. C. M. 29; and 1247 = 29 x 43, = .. the L. C. M. 29 x 43 x 61=1247 x 61=76,067. = 243. From this process it will be seen that: The L. C. M. of two numbers may be found by dividing one of the numbers by their G. C. M. and multiplying the quotient by the other number. The L. C. M. of two prime numbers, or of two numbers prime to each other, is their product. CHAPTER XI. COMMON FRACTIONS. 244. Instead of dividing the unit into 10 equal parts, 100 equal parts, and so on, it is often more convenient to divide it into halves, thirds, quarters, or other equal parts. These are called common fractions. 245. If a line AB be divided into 5 equal parts, at the points C, D, E, and F, the parts AC, AD, AE, and AF are fractions of the whole, being respectively one-fifth, twofifths, three-fifths, four-fifths of the whole line, and are written,,&, f. I. 4 of the parts when 1 unit has been divided into 5 equal parts. II. of 4 units; for if four units be divided into 5 equal parts, one of these parts will be four times as great as one of the parts obtained by dividing one unit into 5 equal parts; that is, will be equal to of one unit. III. The quotient of 4 divided by 5. 247. In the fraction, the lower figure shows into how many equal parts the whole has been divided, and is therefore a divisor. But since it shows the number of parts into which the whole has been divided, it shows the name of each part, and is therefore called the denominator. The upper figure shows how many of these parts are taken, and is therefore called the numerator.* 248. It will be observed that a figure written above the line serves a very different purpose from that of a figure written below the line. A figure written above the line denotes number, a figure below the line, name. 249. The expressions 1, 3, 4, †, 1, 4, 11, are read onehalf, two-thirds, three-fourths, three-fifths, one-sixth, foursevenths, seventeen twenty-firsts. In like manner, all other fractions are read by reading first the numerator, and then the denominator as an ordinal, with an s added if the numerator is more than one. Read: 13, 10, 14, 1880, 21, 8, 12, 14, 4, 17, 21, 1, 2, . 18 ૐ, ૐૐ, 55 19 250. The numerator and denominator are called the terms of a fraction. 251. A proper fraction is one of which the numerator is less than the denominator; as . 252. An improper fraction is one of which the numerator equals or exceeds the denominator; as 7, 15. It is obvious that when the numerator is greater than the denominator, more than one unit must be regarded as divided into parts; thus, means that three units have been divided each into thirteen equal parts, and that all the parts of two units and three parts of the third are taken. 253. A mixed number is an expression consisting of a whole number and a fraction; as 54. It means that some entire units are taken, and a fraction of another unit. *Numerator and denominator are derived from the Latin numerare, to count, and denominare, to name. |