tion till there is no remainder, and the last divisor will be the greatest common measure. To ascertain the greatest common measure of three or more numbers, find the greatest common measure of any two of them: then that of this greatest common measure and another of them: and so on to the last. Examples. (1) Required the greatest common measure of 428571 and 999999. Here, 4 2 8 571) 999 999 (2 8 5 7 14 2 1 4 2 8 5 7 ) 4 2 8 5 7 1 (3 therefore 142857, being the last divisor, is the greatest common measure. (2) What is the greatest common measure of the numbers 12, 42 and 63 ? Here, by inspection, 6 is the greatest common measure of 12 and 42; and to find that of 6 and 63, we have 6) 6 3 (10 60 3)6(2 6 therefore, 3 is the common measure required. (3) Determine the greatest factor common to 741, 1131, 1183 and 1989. Proceeding by the directions of the rule, we have 741)1131 (1 741 390)741(1 351) 390(1 351 39) 351(9 or, 39 is the greatest common measure of 741 and 1131 : and to find that of 39 and 1183, the process will be 39) 118 33 0 117 13) 39(3 39 or, 13 is the greatest common measure of 741, 1131 and 1183: whence, 13 is the factor to be determined, since it also divides 1989 without a remainder. In the second example, it is immaterial in what order the numbers are taken; and in the last instance it will be found that the number required is the greatest common measure of the common measures of every two of them that can be selected. 53. The following remarks will be of service in making use of the last rule. If the figure in the units' place be divisible by 2, the number is divisible by 2. If the figures in the units' and tens' places be 4, or be divisible by 4, the number is divisible by 4. If the figures in the units', tens' and hundreds' places be 8, the number is divisible by 8. If the sum of all the figures be divisible by 3 or 9, the number is divisible by 3 or 9. If the figure in the units' place be 5 or 0, the number is divisible by 5. If the sums of the alternate figures beginning at either end be equal, or one sum exceed the other by 11, or by any multiple of it, the number is divisible by 11. 54. To find the least common multiple of two numbers. To find the least common multiple of 18 and 30, we observe that so that the least number which contains them both exactly is evidently 6 × 3 × 590, or the product of 18 and 30 divided by 6 their greatest common measure: and hence we have the following Rule. Rule for finding the Least Common Multiple. Multiply either of the numbers by the quotient arising from dividing the other by the greatest common measure, and the product will be their least common multiple. If there be more than two numbers, proceed in the same way with the least common multiple of any two of them and the third: and so on, till they are all taken. Examples. (1) What is the least common multiple of 209 and 304? Here, we have the operations below: 209)304 (1 209 95)209(2 190 19)95(5 = so that 19 is the greatest common measure of the numbers proposed: and the number required will therefore(209 × 304) ÷ 19 = (209 ÷ 19) × 304 = 11 × 304, (304 × 209) 19 (304 ÷ 19) × 209 = 16 × 209: both of which being multiplied out, amount to 3344. (2) Determine the least common multiple of 64, 250 and 432. or, = = The greatest common measures of 64 and 250 is 2, and their least common multiple is 8000: the greatest common measure of 8000 and 432 is 16, and the least common multiple of the three numbers proposed will therefore be 216000: and, as in the preceding Article, the order in which the numbers are taken will have no influence upon the result. 55. When the least common multiple of several numbers is required, and no measures common to any two or more of them appear at first sight to exceed 12, the easiest method is to place the numbers in a row, and to divide such of them, as admit of it, by the primes; or prime numbers 2, 3, 5, 7, 11, repeated when it can be done, as often as possible: then, the product of all these divisors and the numbers in the last line, will be the least common multiple. Thus, to find the least common multiple of 2, 3, 8, 9, 15, 21 and 35, we shall have the following scheme: 2) 2, 3, 8, 9, 15, 21, 35 and the least common multiple is 2 × 2 × 3 × 5 x 7 x 2 x 3 = 2520. This Rule is founded upon Article (51); very little attention is required to see the reason of it, and the following examples will furnish its practice. Examples for practice. (1) 3, 5, 9. (2) 8, 9, 12, 18. (3) 6, 15, 27, 35. (4) 3, 9, 7, 15, 28, 42. (5) 8, 18, 28, 36, 54, 72, 90. General proofs of all that has been said here, may be found in the Author's Elements of Algebra. CHAPTER II. APPLICATION OF ARITHMETIC ΤΟ NUMERICAL MAGNITUDES OF VARIOUS DENOMINATIONS, NOT CONNECTED BY THE 56. In the preceding Chapter we have considered only such abstract numbers as are formed by figures whose local values are always regulated by the same fixed number ten: but the rules given are easily extended to concrete magnitudes wherein the local values of the figures are connected by more numbers than one; as for instance, to Pounds, Shillings, Pence, and Farthings, where four farthings are equivalent to one penny, which is the next higher denomination; twelve pence to one shilling, which is the next denomination in order; and twenty shillings to one pound: the different numbers 4, 12 and 20 connecting the denominations, in the same manner as the fixed number 10, was supposed to connect the denominations of Integers. The processes employed in cases of this nature are Reduction, and the fundamental operations then called Compound Addition, Compound Subtraction, Compound Multiplication and Compound Division, each of which will be exemplified in order: and the Tables by means of which they are conducted, will be found at the beginning of the work. REDUCTION. 57. DEF. Reduction is the converting or changing of numerical quantities, from one or more denominations to one or more others, such that the real or absolute values shall remain unaltered: and its operations will evidently depend upon the principles already explained. Ex. Reduce £25. 13s. 63d. into farthings; and perform the converse operation. The correctness of the following operations will be |