the new divisor and quotient, would increase the product as much as the number divided. The same might be shown as it respects any other numbers, for the last product would always be made up of each of the given numbers a certain number of times. 2. What is the least common multiple of 20, 6, 8, 15, 7 ? 5) 20 6 8 15 7 4) 4 6 8 3 7 5 x 4 = 20 × 3=60X2= 120 × 7=840 Ans. 3. What is the least common multiple of 2,3,7,9 % - - , Ans. 126. 4. What is the least number that can be divided by 7, 11, or 13? Ans. 1001. 5. What is the least common multiple of 26, 100, 234, 1050? Ans. 81900. - - PROBLEM II. To find the greatest common divisor, or measure of two numbers. RULE. Divide the greater number by the less, and the divisor by the remainder. Continue to divide the last divisor by the last remainder until nothing remains; the last divisor will be the greatest common divisor, Or measure. 1. What is the greatest common divisor, or measure of 125 and 2225? . 2 3 5 0 sum of the subtrahends and last divisor, or remainder. . . Illustration.—In the operation of the last question, 25 the last divisor or remainder, will divide itself; it also divides the last dividend without a remainder. But the last divisor and the last dividend added together make 125, the least given number, which is 5 times 25, consequently 25 will divide it without a remainder. All the subtrahends are made by multiplying 25, or numbers made up of 25's, therefore all the subtrahends are divisible by 25. It will be seen by the operation, that the sum of the subtrahends and last divisor, equals the sum of the given numbers; the given numbers therefore are composed of 25's, and can be divided by 25 without remainders. By examining the operations of other questions, it will be seen that all the subtrahends will be composed of the last divisor, or some multiple of the last divisor, a certain number of times. The sum of the subtrahends and last divisor will always equal the given numbers, for making the divisor the next divi dend, we always subtract from some part of the given numbers, therefore the last divisor will always divide the given numbers without a remainder. 2. What is the greatest common divisor of 24 and 1607 - Ans. 8. 3. Required the greatest common measure of ####. - Ans. 27. 4. What is the greatest common measure of forg ? Ans. 2. 5. Find the greatest number that will divide both parts of the fraction roor without a remainder. Ans. 1. 6. If I have two quantities of coal, one 468 bushels, the other 612, how many bushels must the greatest measure contain that will exactly measure both quantities” . Ans. 36. REDUCTION OF FRACTIONS. * ~ *- : Reduction of fractions is changing the form of fractions without altering their values. The same part or parts of an unit may be expressed by different fractions. . We can only tell what part of an unit a fraction is, by comparing the numerator with the denominator. The fractions 3, #, #, #, all express the same part of an unit, for in each the numerator is half the denominator, and each denominator being equal to an unit, the value of each fraction is 4 of an unit. By dividing both terms of the three last fractions by 2, 3, and 4, respectively, we shall obtain # for the quotient of each fraction. When both terms of the fraction are divided by the same number, the quotients will be a fraction expressing the same part or parts of an unit as the numbers divided, for they are both lessened by the same number. If we divide both parts of ; by 2, 2) # (;, we take half the numerator for a new numerator, and half the denominator for a new denominator. One half of a number must be the same part of a half of another number, that one whole number is of the other. 2 is half of 4, and 1 is half of 2, so that i is the same part of 2, that 2 is of 4, that is, 4, is the same part of a unit as #. If we divide both parts of a fraction by any other number, the quotients will be a fraction expressing the same value as the given fraction, for dividing by the same number, we take the same part of the numerator that we do of the denominator; thus, 6) +}=#;—here we take a sixth part of the numerator, and a sixth part of the denominator, and find 2 to be the same part of 3, that 12 is of 18. JNote 1.—When no number greater than 1 will divide both terms of the fraction without a remainder, the fraction is said to be in its lowest terms. * * * * * From the above reasoning we obtain the truth of the following RULE, For reducing a fraction to its lowest terms. Find the greatest common measure, and divide both terms of the fraction by it, the quotients will be a fraction expressing the same value as the given fraction. JNote 2.—When the terms of the fraction are small numbers, we can generally discover at sight what number will divide both without a remainder, if the fraction be not then in its lowest terms, we must continue dividing in a similar manner until it is." 1. Reduce for to its lowest terms. Operation. To find the greatest Given nuCommon measure, merator. 2 3 5) 1 1 7 5 ( 5 2 3 5 ) 2 3 5 (1 new nu1 1 7 5. 2 3' 5 merator. * If the two right hand figures of the fraction be even numbers, the terms of the fraction are divisible by 2; if 5's, divisible by 5.5 if ciphers, divisible by 10, or by 5. If the sum of the figures in each term be divisible by 3 or 9, both terms are divisible by 3 or by 9. The fraction in its lowest terms is #. 2. Reduce ### to its lowest terms. - * -> * * ... operation as directed in the Note, so *** * --- Ans, or, 6. In or, what part of the denominator is the numerator 7 Ans. }. To change a mixed number to the form of a fraction, that is, to an improper fraction. RULE, Multiply the whole number by the denominator, and to the product add the numerator; the sum placed over the denominator will form an improper fraction, equal in value to the mixed number. 1. What improper fraction is equal to 104 ° 10x5+1 = **, the answer. Illustration.—The denominator being equal to an unit in the whole number, (see explanation of fractions, page 130.) if it be placed in the numerator as many times as there are units in the whole number, the numerator will contain as many units as the whole number, and by adding the given numerator, the sum will equal the whole number and the given fraction. In multiplying the whole number by the denominator, we place the denominator in the product once for each unit in the whole number, or manltipli |