common divisor cannot be greater than the difference between the two numbers, which, in this case is 18. Therefore, we have PROPOSITION VII. The greatest common divisor of two numbers is obtained by dividing the greater by the less, then dividing the divisor by the remainder, and continuing to divide the last divisor by the last remainder until nothing remains. The last divisor will be the greatest common divisor sought. Q. Will the common divisor of two numbers divide their remainder after division? How do you find the greatest common divisor of two numbers? 3. Find the greatest common divisor of the two numbers 63 and 81. 4. Find the greatest common divisor of 315 and 405. Ans. 45. Ans. 5. What is the greatest common divisor of the two numbers 2205 and 2835? 6. Find the greatest common divisor of 1157 and 623, Ans. 7. Find the greatest common divisor of 792 and 1386. Ans. 198. NOTE-If it be required to find the greatest common divisor of more than two numbers, find first the greatest common divisor of two of them, then of that common divisor and one of the remaining numbers, and so on, for all the numbers: the last common divisor will be the greatest common divisor of all the numbers. Ans. 8. What is the greatest common divisor of 246, 372, and 522? 9. What is the greatest common divisor of 492, 744 and 1044? Ans. 12, LEAST COMMON MULTIPLE. § 87. A number is said to be a common multiple of two or more numbers, when it can be divided by each of them without a remainder. For example, 6 is a common multiple of 2 and 3, because it is exactly divisible by each of them. So likewise, 12 is a common multiple of 2, 3, 4, and 6, because it is divisible by each of them. The least common multiple of two or more numbers, is the least number which they will separately divide without a remainder. For example, 12 is a common multiple of 2 and 3, but it is not the least common multiple, since 6 is also divisible by 2 and 3. Now 6 being the least number which is so divisible, it is the least common multiple of 2 and 3. To find the least common multiple of several numbers, we have the following RULE. I. Place the numbers on the same line, and divide by the least number that will divide two or more of them without a remainder, and set down in a line below the quotients and the undivided numbers. II. Divide as before, until there is no number greater than 1 that will exactly divide any two of the numbers: then multiply together the numbers of the lower line, and the divisors, and the product will be the least common multiple. If, in comparing the numbers together we find no common divitheir product is the least common multiple. sor, EXAMPLES. 1. Find the least common multiple of 3, 4 and 8. We first see, that 2 will divide 4 and 8, but as it will not divide 3, we bring down 3 into the 2nd line: we again see that 2 is a common divisor of 2 and 4; and as there is no com OPERATION. 2)3.. 4...8 4 3 1 2 Ans. 2x1×3×2×2=24. mon divisor between any two of the numbers of the last line, it follows that 2×1×3 multiplied by the two divisors, is the least common multiple. Q. When is one number said to be a common multiple of two or more numbers? Of what numbers is 6 a common multiple? Of what numbers is 8 a common multiple? What is the least common multiple of two or more numbers ? What is the difference between a common multiple, and the least common multiple? Give the rule for finding the least common multiple. If the numbers have no common divisor what is the least common multiple? OPERATION. 3)3...8...9 3 8 2. Find the least common multiple of 3, 8, and 9. We arrange the numbers in a line and see that 3 will divide two of them. We then write down the quotients 1, and 3, and also the 8 which cannot be divided. Then as there is no common divisor between any two of the numbers, 1, 8, and 3, it follows that their product, multiplied by the divisor 3, will give the least common multiple sought. 1x8x3x3=72. 3. Find the least common multiple of 6, 7, 8 and 10. Ans. 4. Find the least common multiple of 21 and 49. 5. Find the least common multiple of 2, 7, 5, 6. Find the least common multiple of 4, 14, 7. Find the least common multiple of 13 and Ans. 147. 6 and 8. Ans. 840. 28 and 98. Ans. 8. Find the least common multiple of 12, 4 and 7. 9. Find the least common multiple of 6, 9, 4, 10. Find the least common multiple of 13, 12 Ans. 84. 14 and 16. Ans. 1008. and 4. Ans. 11. What is the least common multiple of 11, 17, 19, 21 and 7? Ans. 74613. REDUCTION OF VULGAR FRACTIONS. § 88. Reduction of Vulgar Fractions is the method of changing their forms without altering their value. A fraction is said to be in its lowest terms, when there is no number greater than 1 that will divide the numerator and denominator without a remainder. Q. What is reduction? When is a fraction said to be in its lowest terms? Is one-half in its lowest terms? Is two-fourths? Is threefourths? CASE I. § 89. To reduce an improper fraction to its equivalent whole or mixed number. RULE. Divide the numerator by the denominator, the quotient will be the whole number; and the remainder, if there be one, placed over the given denominator will form the fractional part. EXAMPLES. 1. Reduce 84 and 67 to their equivalent whole or mixed numbers. OPERATION. 4)84 Ans. 21 OPERATION. 9)67 Ans. 76/ 9 It was shown in § 44, that the value of every fraction is equal to the quotient arising from dividing the numerator by the denominator: hence the value of the fraction is not changed by the reduction. Q. How do you reduce a fraction to its equivalent whole or mixed number? Does this reduction alter its value? Why not? What is four-halves equal to ? Eight-fourths? Sixteen-eighths? Twenty-fifths? Thirty-six-sixths? Four-thirds? What is nine-fourths equal to ? Four-fifths? Seventeen-sixths? Eighteen-sevenths? 2. Reduce to a whole or mixed number? Ans. 123. 3. In of yards of cloth, how many yards? yd. Ans. 4. In 51 of bushels, how many bushels? 5. If I give of an apple to each one of 15 children, how many apples do I give? Ans. 5. 6. Reduce 327 3672 50287 987625, to their whole or mixed numbers. 1259 153 6941 723019 7. If I distribute 878 quarter apples among a number of bovs, how many whole apples do I use? Ans. CASE II. § 90. To reduce a mixed number to its equivalent mproper fraction. RULE. Multiply the whole number by the denominator of the fraction; to the product add the numerator, and place the sum over the given denominator. EXAMPLES. 1. Reduce 4 to its equivalent improper fraction. Here 4x5=20: then 20+4=24; hence, 24 is the equivalent fraction. Ans. 2. This rule is the reverse of Case I. In the example 4 we have the integer number 4 and the fraction. Now 1 whole thing is equal to 5 fifths, and 4 whole things are equal to 20 fifths; to which, add the 4 fifths, and we obtain the 24 fifths. Q. How do you reduce a mixed number to its equivalent improper fraction? How many fourths are there in one? In two? In three? How many sixths in four and one-sixth? In eight and two-sixths? In seven and three-sixths? In nine and five-sixths? In ten and fivesixths? How many eighths in two and one-eighth? In three and three-eighths? In four and four-eighths? In five and six-eighths? In seven and seven-eighths? In eight and seven-eighths? 2. Reduce 47% to its equivalent improper fraction? 1009 Ans. 287. 3. Reduce 67637, 87433, 69047, 36794, to their equivalent improper fractions. Ans. 4. Reduce 8473, 87417, 67426398, to their equiva § 91. To reduce a fraction to its lowest terms. RULE. I. Divide the numerator and denominator by any number that will divide them both without a remainder, and then divide |