that 14, which has before been shown to be a common measure of 42 and 154, must be itself their greatest common measure. To find the greatest common measure of several numbers.Find the G. C. M. of two of the numbers; then of that result and a third number; then of that result and a fourth; and so on. The last G. C. M. is the one required. Ex. Find the G. C. M. of 948, 1185, 1580. 948) 1185 (I 237) 948 (4 And 237) 1580 (6 1422 158) 237 (1 79) 158 (2 158 ... 79 is the G. C. M Exercise 13. (1) Write down all the factors of 48, 90 132, 160. (2) Separate into their prime factors 56, 84, 120, 154, 204, 1001. (3) Write down all the common factors of 24 and 36; of 28 and 70; of 90 and 120: and the G. C. M. of 48 and 88; and of 27, 63, and 120. Find the G. C. M. of (4) 289 and 340. (6) 621 and 805. (8) 817 and 1786. (10) 9789 and 11986. (12) 26467 and 30457. (14) 78, 130, 195. (16) 136, 204, 357, 459. (18) 3864, 3404, 3657. (20) 15561, 11115, 13585. (5) 1000 and 825. (7) 17104 and 27794. (9) 4257 and 10836. (11) 30072 and 133784. (13) 24, 60, 84, 128. (15) 252, 315, 420, 504. (17) 128, 192, 320, 368, 432. (19) 3555, 4977, 6636. 16.-A number is said to be a multiple of another number, when it contains that number an exact number of times; thus 24 is a multiple of 12, 8, &c. A common multiple of two or more numbers is a number which contains each of them exactly; thus 60 is a common multiple of 15 and IO. The least common multiple of two or more numbers is often written for shortness, L. C. M. RULE.- -To find the least common multiple of two or more numbers.-Set down the numbers, and divide by any prime number which is contained in more than one of them, setting down again those numbers which do not contain the divisor exactly. Continue this process until no two of the quotients have a common factor; then the product of the divisors and last quotients will be the L. C. M. required. Ex. (1) Find the L. C. M. of 3, 8, 9, 10, 12. and the L. C. M.2 × 2 × 3 × 2 × 3 × 5=360. This method depends upon the facts: (a) that any number which contains all the prime factors of another, repeated the same number of times, will be a multiple of that other; thus any number which contains two 28 and a 3, will be a multiple of 12. (B) that a number cannot be a multiple of another, unless it contains all the prime factors of that other, repeated the same number of times. For instance, a number not containing 3, or not containing two 2s, cannot be a multiple of 12. Now by the above process, we obtain either as divisors or as last quotients the prime factors of each of the given numbers; therefore the product of the divisors and quotients will contain all those factors, and will therefore, by (a), be a multiple of each of the given numbers. Thus, looking at the factors of which it is composed, 360 contains 2 × 2 × 2 or 8: it contains also 2 x 5 or 10; &c. Again, no numbers occur as divisors or last quotients, except prime factors of at least one of the given numbers; and every such factor occurs only so many times as it is repeated in some one of the given numbers; therefore, by (B), the product will be the least common multiple of the numbers. If either of the given numbers be contained in one of the others, it may be omitted, since any multiple of the latter will be a multiple also of the former. Ex. (2) Find the L. C. M. of 8, 12, 16, 18, 27, 72. 28, 17, 16, 18, 27, 72 Here 8 is contained in 16, and 12 and 18 are each contained in 2 8, 27,36 72. 2 4, 27, 18 27,9 2, And in the fourth line, 9 is contained in 27. ... L. C. M.= 2 X2 X2 X2 X27=432. If the given numbers are large, it may be necessary to obtain the common factors by the process for finding the G. C. M. Ex. (3) Find the L. C. M. of 1426 and 989. ... L. C. M.=23×62 × 43=1426 × 43=61318. Exercise 14. Find the least common multiples of (1) 2, 5, 6, 8, 10. (3) 6, 10, 15, 20. -(5) 9, 10, 12, 25, 30. (2) 3, 8, 12, 16. (4) 7, 9, 12, 21, 28. (7) 6, 8, 10, 12, 14, 16. (9) 20, 25, 30, 35, 40 60. (11) 48, 54, 81, 144, 162. (13) 7, 8, 11, 13, 15. (15) 221 and 351. (17) 195, 234, 225. (8) 15, 16, 18, 20, 22, 24. (10) 56, 64, 70, 84, 112. (12) 75, 100, 120, 150, 180. (14) 112, 168, 196, 224. (16) 365 and 511. (18) 194, 291, and 216. (19) What is the least number which contains exactly each of the first twelve natural numbers? (20) Two bells toll at intervals of 6 and 8 seconds respectively, commencing at the same instant; at what intervals will they toll simultaneously? 17.-A power of a number is the product of the number multiplied by itself any number of times. The number multiplied once by itself (i.e. two such numbers multiplied together) is called the second power, or square: when multiplied twice by itself (i. e. three such numbers multiplied together) the product is called the third power, or cube: so the product of four such numbers is called the fourth power; &c. Thus of 5; 25, 125, 625, 3125, &c., are the square, cube, fourth power, fifth power, &c. A root of a number is the converse of a power, and is such a number as being multiplied by itself once, twice, &c., will produce the one given. Thus the square root of 49 is 7, because the square of 7 is 49; the cube root of 125 is 5, because the cube of 5 is 125; so the fourth root of 81 is 3; the fifth root of 32 is 2, &c. A power of a number is indicated by a small figure, called an index, placed to the right of the number, and a little above it, thus 52, 53, 54, &c. A root is indicated by a radical, with a small figure to the left, thus 2 √125. √49, 125, 81, &c. If no figure is expressed, the square root is intended; as √49. CHAPTER IV. VULGAR FRACTIONS. 18.—IN Vulgar Fractions, a unit or whole thing is considered to be divided into an exact number of equal parts, of which a certain number are taken. с D E F B For instance if a line AB be divided into five equal parts in the points C, D, E, F the portions AC, AD, AE, AF, are fractions of the whole, being respectively one-fifth, two-fifths, three-fifths, four-fifths of it. Again, a pound being divided into 20 shillings, a shilling is one-twentieth of a pound, nine shillings are ninetwentieths, &c. A fraction is expressed by two numbers written one above the other, with a line between them, as &; the lower number, which is called the denominator, indicating the number of parts into which the unit was divided; while the upper one, or numerator, indicates the number of these parts which are taken. Thus is the quantity obtained by dividing a unit into eight equal parts, and taking three of them. So, in the case taken above, AC, AD, AE, AF, will be written respectively as,,,, of the whole line AB. Exercise 15. (1) What fraction is one foot of a yard? and ten inches of a foot? (2) Express 3s. as a fraction of a pound; and 16s. as a fraction of a guinea. (3) Express 7d. and 4s. 7d. as fractions of a pound. Express 7 furlongs as a fraction of a mile, 27 minutes as a fraction of an hour, and 3 rds. 8 ples. as a fraction of an acre. |