The 3 being common to the 9 is omitted, and the numbers left, 7, 9, 5, and 11, are multiplied into the 72, and the L. C. M. is found. EXERCISES.-XXIV. Find the least common multiple of the following quantities: (1) 42, 21, 14, 7, 84, 12, 63. (5) 1, 2, 3, 4, 5, 6, 7, 8, 9. 9. Find the least common multiple of 225, 255, 289, 1023, and 4095. 10. Find the least number which is divisible by 7, 14, 21, 35, and 90 separately without a remainder. 11. I have the following numbers to reduce to a common denominator 9, 81, 27, 3, 243, 567; find the least common denominator. : 12. Find the prime factors of 111540, 42336, and 67392, and thence write down the smallest number which they will all divide without a remainder. 13. Find the least common multiple of 4, 5, 6, 10, 12, and 180. Greatest Common Measure.-The highest common factor of two or more numbers is termed their greatest common measure, written G. C. M. Frequently the G. C. M. can be determined by an inspection of the numbers. Find the greatest common measure of 3891 and 8577. In this example the G. C. M. is 3, it is the last divisor. We begin by dividing the greater number by the less, then the remainder left is made the divisor, and the previous divisor becomes the dividend. This is continued until we have no remainder, the last divisor being the G. C. M. The first remainder is 795, then 3891 becomes the second dividend, next the remainder 711 becomes the divisor, and the dividend 795, and so on until we have no remainder; the last divisor 3 is the G. C. M. As a second example, find the G. C. M. of 13515 and 13787. 13515)13787(1 272)13515(49 1088 13515 13787 13515 49 27213515 1088 85 On the right-hand side is given a second method of putting down the operations; at first it is not so easy as the method adopted on the left hand of the page, but when the G. C. M. of three or four quantities is required, then the method possesses compactness and neatness; but the figures are exactly the same in quantity as the ordinary method. The G. C. M. is chiefly employed in reducing fractions to their lowest terms, while the L. C. M. is essentially necessary when fractions have to be reduced to a common denominator; both of these subjects will be presently brought under consideration. Find the G. C. M. of the following quantities : 9. Find the G. C. M. of 68635 and 19721, and the L. C. M. of 8, 9, 10, 12. 10. Find the L. C. M. of 49, 77, 143, 91, 1001, and state the rule in words. 11. Find the L. C. M. of 40, 42, 44, 48. 12. Find the greatest number that will divide 2175 and 91125 separately without a remainder. 13. Suppose that three men can do a piece of work, in 405, 420, 450 days respectively, express the relative values of the works of the three men in the simplest manner possible. FRACTIONS. A Fraction is a part of a whole. A fraction is usually expressed by means of two numbers, one placed over the other with a line between them, as,. The first fraction, †, means this, that the whole number is divided into 9 parts, and the one, or upper figure, indicates that 1 alone of the 9 parts is under consideration. The second fraction, , means that the whole number is divided into 11 parts; this is indicated by the bottom figure; the top figure, 7, shows that seven of these parts are taken out of the 11. Numerator.-Of any given fraction the upper figure is termed the numerator, as,,, etc., the 1, 3, and 5 are called the numerators, as they enumerate or tell us how many parts of the whole are taken. Denominator. Of any given fraction the lower figure is called the denominator, because it is the name of the fraction, and tells us of what denomination it is, while the numerator shows how many of the given denomination are taken-,,, the 2, 4, and 6 are the denominators. Fractions are of several classes, proper, improper, simple, mixed, complex, and compound. A Proper Fraction.—A proper fraction is one that has a numerator less than its denominator-,, 1, §, 1, 11, are all proper fractions. An Improper Fraction.-An improper fraction is one that has a numerator greater than its denominator-, †, V, H, H, are all improper fractions. A Simple Fraction is one whose numerator and denominator are both integral numbers, as §,, and . The fraction 23 is a proper fraction; it is also a compound fraction, of which we shall speak in a moment; but at the same time 2 is not a simple fraction, although it may be reduced to one; only such fractions as, 1, 18, etc., are simple. A Mixed Number, or Mixed Fraction, as it is sometimes improperly called, consists of a whole number and a fraction, as 21, 7, 21, etc. A Complex Fraction has its numerator or denominator, or both, in the form of a mixed fraction, as 216 17 18 7488 3 17 17% etc. -" A Compound Fraction.-A compound fraction is a fraction of a fraction, or a part of a part, as of; 2 of §; 43 of 71, 21 41 etc. All these are called Vulgar Fractions, which simply means common fractions, to distinguish them from decimal fractions. In vulgar fractions it will be observed that when the numerator is less than the denominator, the fraction is less than the whole number or unity, and when greater, greater. Fractional Forms and Mixed Numbers.-First it will be shown how mixed numbers are reduced to improper fractions or into a complete fractional form, next how to reduce any improper fraction to a whole or mixed number. To reduce a mixed number to an improper fraction.-Multiply the whole number by the denominator of the fraction, and add in the numerator; this gives the numerator of the new fraction; the denominator is the same as the original fraction which formed part of the mixed number. Reduce 18 and 7% and 14§ to improper fractions. 18; 78; 14=117. (1.) We say 11 times 18 are 198 and 5 make 203; this is the numerator; the denominator is the original 11. (2.) 9 times 7 are 63 and 2 are 65; this is the numerator; the denominator is as before 9. (3.) 8 times 14 are 112 and 5 are 117, the numerator; the 8 remains as the denominator. To reduce an improper fraction to a whole or mixed number.This is the reverse of the last. Divide the numerator by the denominator, and set down the number of times that the one goes into the other as the whole number, the remainder is the numerator of the factors; the denominator remains unchanged. Reduce 112, 2487, and to mixed fractions. 17 112=12; 2487=146; 9=4713. (1.) We say 9 into 112 goes 12 times, this is the whole number; there is a remainder 4, this is the numerator, while the denominator 9 remains. (2.) 17 into 2487 goes 146 times with a remainder 5; the 146 is the whole number, the 5 is the numerator of the fraction, with 17 for the denominator. (3.) 19 into 910 goes 47 times with 17 over; the 47 is the whole number, and the 17 the numerator of the new fraction 471%. To reduce fractions of two or more ratios to simple fractions.— If any terms in the top will cancel those in the bottom, omit them; or if there be any factors common to the numerators and denominators of the fractions, omit them. Reduce of, of %, and of to simple fractions. (1.) In the first, 5 being common to 5 and 35 it is cancelled; while 4 being common to 8 and 12, that is also cancelled; then the 3 left, or really 3× 1, forms the numerator, while the denominator is the product of 2 and 7=14. (2.) In the second 3 is common to the 3 and 9, and 4 to the |