86. 41×67-89÷41×3×3=what? 87.56×7÷46×867-82-what? 88. 916×4×3÷7÷976×8756=what? 93. 3×2÷(4×3÷3)=3x2÷4÷3×3=what? 95. 41×2×(6÷7×3÷4)=41×2×6÷7×3÷4=what? 97. (4÷2×3)÷(2×4)=4÷2×3÷2÷4=what? 99. 4÷6×7÷8x9÷7x8÷(9÷2)=what? 107. 59×60 33x46x25÷÷÷75×2=what? QUESTIONS.-If the divisor is increased in any ratio, how will the quotient be affected? (66., h.) If the divisor is decreased in any ratio, how will the quotient be affected? (66., i.) If the divisor is any number of times too small, how will the quotient be affected? If the divisor is any number of times too large, how will the quotient be affected? If the dividend is increased or diminished in any ratio, how will the quotient be affected? (66., j.) (66., k.) The product of what two numbers plus the remainder is equal to the dividend? (66., 7.) What number divided by the multiplier will equal the mul tiplicand? (66., m.) When the sum and difference of two numbers are given, how are the numbers found? (66., n.) LESSON XI. 118. A Common Divisor of two or more numbers any number greater than 1, that will divide each of them without a remainder. is 119. The Greatest Common Divisor of two or more numbers is the greatest number that will divide them without a remainder. (a.) What is the greatest common divisor of 84 and 132? (b.) What is the greatest common divisor of 84 and 203? ANALYSIS.-(a.) By resolving the numbers into their prime factors, we find that those of 84 are 2, 2, 3, 7; and those of 132 are 2, 2, 3, 11. NOTE. For the benefit of those who prefer this method, we give the analysis of it here. It depends upon the following principles: I. A divisor of any number is a divisor of any multiple of that number. II. A divisor of any two numbers is a divisor of their sum. III. A divisor of any two numbers is a divisor of their difference. Mirst ascertain whether 84 is a divisor of 203, if so, it is the divisor sought. As any divisor of 84 is a divisor of 168 (Prin. I.; therefore, any divisor of 84 and 203 is also a divisor of 203 and 168. As any divisor of 203 and 168 is a divisor of 35 (Prin. III.) therefore, any divisor of 84 and 203 is a divisor of 84 and 35. Again, as any divisor of 84 is a divisor of 168 (Prin. I.) therefore, any divisor of 84 and 35 is a divisor of 168 and 35; but any divisor of 168 and 35 is also a divisor of 203: (Prin. II.) consequently, any divisor of 84 and 35 is a divisor of 84 and 203; and hence, the greatest common divisor of 35 and 84 is the greatest common divisor of 84 and 203 In the same manner the greatest common divisor of 14 and 35 is found to be the greatest common divisor of 35 and 84; also, the greatest common divisor of 7 and 14 is found to be the greatest common divisor of 14 and 35. As 7 is the greatest common divisor of itself and 14. it is, therefore, the greatest common divisor of 84 and 203. 5 By inspection, we find that the factors 2, 2, 3, are common to both 84 and 132. Since only common factors will divide both numbers, the product of all the common factors must be the greatest factor that will divide both without a remainder. Twelve is the product of the common factors 2, 2, 3, and is, therefore, the greatest common divisor of 84 and 132. I. RULE.-(a). Resolve the numbers into their prime · factors, select the common factors, and their product will be the greatest common divisor required. II. RULE.-(b.) Divide the greater number by the less, and, if there be a remainder, divide the preceding divisor by it, and so continue dividing, until there is no remainder. The last divisor will be the greatest common divisor sought. If there are more than two numbers, find the greatest common divisor of the first two, then of the divisor thus found and one of the remaining numbers, and so continue until the greatest common divisor of all the given numbers is found. III. RULE.-(c.) Divide the numbers by any number that will divide all of them without a remainder; continue dividing each successive set of quotients until they have no common factor: the product of the divisors will be the greatest common divisor required. LESSON XII. 110. What is the greatest common divisor of 85 and 95? 111. What is the greatest common divisor of 72 and 168? 112. What is the greatest common divisor of 119 and 121? 113. What is the greatest common divisor of 12, 18, 24, and 30? 114. What is the greatest common divisor of 14, 28, and 21? 115. What is the greatest common divisor of 20, 16, and 48? 116. What is the greatest common divisor of 28, 16, 12, and 8? 117. What is the greatest common divisor of 11, 88, and 99 ? 118. What is the greatest common divisor of 28, 63, 47, and 93? 119. I have three rooms respectively 15, 18, and 24 feet in width; what must be the width of the widest oil-cloth that will exactly fit without cutting? QUESTIONS. How does annexing a cipher to a number affect it? (67., a.) Why? Illustrate the contracted method of dividing by 10, 100, &c. (67., f.) What is United States money? (68.) What are the Gold coins? (69., a.) Silver? (69., b.) Nickel? (69., c.) LESSON XIII. COMMON MULTIPLE. 120. A Multiple of a number is a number that can be divided by it without a remainder. 121. A Common Multiple of two or more numbers, is a number that can be divided by each of them without a remainder. 122. The Least Common Multiple of two or more numbers, is the least number that can be divided by each of them without a remainder. What is the least common multiple of 6, 9, and 12? *NOTE.-Reject the 6, for it is a factor of 12. Factors can be rejected before any subsequent division as well as the first. ANALYSIS. (a.) 1. The least number divisible by 6 must contain only the highest powers of all the different factors of 6, which are 2 and 3. 2. The least number divisible by 6 and 9 must contain only the highest powers of all the different factors of 6 and 9, which are 32 and 2. We reject the first power of 3 found in the given number 6, because we have a higher power of 3 in the given number 9. 3. The least number divisible by 6, 9, and 12, must contain only the highest powers of all the different factors of 6, 9, and 12, which are 22, 32. We reject the first power of 2 in the number 6, and the first power of 3 in the number 12, because we find higher powers of each factor in the numbers 9, and 12. Therefore, 22x32, 36, is the least common multiple of 6, 9, and 12. (b.) This abbreviated method of operation is generally preferred, and the analysis being essentially the same as that of the method preceding, it will not be repeated. RULE.-I. (a.) Resolve the numbers into their prime factors. The product of the highest powers of all the different factors will be the least common multiple. RULE.-II. (6.) Arrange the numbers on a horizontal line, rejecting all that are factors of any of the other numbers; divide by the least prime number that will divide more than one of them without a remainder, and write the quotients and the undivided numbers in a line beneath. Reject factors and continue to divide as before, until there is no prime number greater than a unit that will divide two or more of them without a remainder. The product of the divisors and the undivided numbers is the least common multiple required. LESSON XIV. EXERCISES FOR PRACTICE. Find the least common multiple of each of the following series of numbers. |