124. To find the greatest common divisor. Ex. 1. What is the greatest common divisor or measure of 84 and 132? Ans. 12. FIRST OPERATION. = = = = · Resolving the numbers into their 84 2 X2 X3 X 7 prime factors (Art. 114), thus, 84 2 X 2 X 3 X 7, and 132 132 2 X2 X3 X 11 2 X 2 × 3 × 11, we find the factors 2 X 2 2 X2 X3 = 12. X 3 are common to both. Since only these common factors, or the product of two or more of such factors, will exactly divide both numbers, it follows that the product of all their common prime factors must be the greatest factor that will exactly divide both of them. Therefore 2 × 2 × 3 12 is the greatest common divisor required. = The same result may be obtained by a sort of trial process, as by the second operation. SECOND OPERATION. 84) 132 (1 84 48) 84 (1 36) 48 (1 RULE 2. 12) 36 (3 We know a common divisor of 48 and 84 will also be a divisor of 132 (Art. 122). We next try to find that divisor. It cannot be greater than 48. But 48 will not exactly divide 84, there being a remainder, 36; therefore 48 is not the greatest common divisor. Again, as the common divisor of 36 and 48 will also be a divisor of 84 (Art. 122), we try to find that divisor, knowing that it cannot be greater than 36. But 36 will not exactly divide 48, there being a remainder, 12; therefore 36 is not the greatest common divisor. Since 84 cannot be exactly divided by a number greater than itself, if it will also exactly divide 132, it will be the greatest common divisor sought. But, on trial, we find 84 will not exactly divide 132, there being a remainder, 48. Therefore 84 is not a common divisor of the two numbers. As before, the common divisor of 12 and 36 will be a divisor of 48 (Art. 122); we make a trial to find that divisor, knowing that it cannot be greater than 12, and find 12 will exactly divide 36. Therefore 12 is the greatest common divisor required. RULE 1. Resolve the given numbers into their prime factors. The product of all the factors common to the several numbers will be the greatest common divisor. Or, Divide the greater number by the less, and if there be a 124. What are the rules for finding the greatest common divisor of two or more numbers? remainder divide the preceding divisor by it, and so continue dividing until nothing remains. The last divisor will be the greatest common divisor. NOTE.- When the greatest common divisor is required of more than two numbers, find it of two of them, and then of that common divisor and of one of the other numbers, and so on for all the given numbers. Another method is to divide the numbers by any factor common to them all; and so continue to divide till there are no longer any common factors; and the product of all the common factors will be the greatest common divisor required. EXAMPLES FOR PRACTICE. 2. What is the greatest common divisor of 85 and 95? Ans. 5. 3. What is the greatest common divisor of 72 and 168? Ans. 24. 4. What is the greatest common divisor of 119 and 121? Ans. 1. 5. What is the greatest common divisor of 12, 18, 24, and 30? Ans. 6. 6. Having three rooms, the first 12 feet wide, the second 15 feet, and the third 18 feet, I wish to purchase a roll of the widest carpeting that will exactly fit each room without any cutting as to width. How wide must it be? Ans. 3 feet. A COMMON MULTIPLE. 125. A Multiple of a number is a number that can be divided by it without a remainder; thus 6 is a multiple of 3. 126. A Common Multiple of two or more numbers is a number that can be divided by each of them without a remainder; thus 12 is a common multiple of 3 and 4. 127. The Least Common Multiple of two or more numbers is the least number that can be divided by each of them without a remainder; thus 30 is the least common multiple of 10 and 15. NOTE. -A multiple of a number contains all the prime factors of that number; and the common multiple of two or more numbers contains all the prime factors of each of the numbers. Therefore, the least common multiple of two or more numbers must be the least number that will contain all the prime factors of them, and none others. Hence it will have each prime factor taken only the greatest number of times it is found in any of the several numbers. 125. What is a multiple of a number?-127. The least common multiple of two or more numbers ? 128. To find the least common multiple. Ex. 1. What is the least common multiple of 6, 9, 12? Ans. 36. FIRST OPERATION. 2 X 3 3 X 3 2 X 2 X 3 6 9 12 = = Resolving the numbers into their prime factors, us, (6 2 X 3, and 9 = 3 X 3, and 12 2 X 2 X 3. we find their different prime factors to Le 2 and 3. The greatest number of times the 2 occurs as a factor in any of the numbers is twice, as 2 X 2 X3 X 3 2 X 2 in 12; and the greatest number of times the 3 occurs in any of the numbers is also twice, as 3 X 3 in 9. Hence 2 X2 X3 X 3 must be all the prime factors that are necessary in composing 6, 9, and 12; and, consequently, the product of these factors nust be the least number that can be exactly divided by 6, 9, and 12. Therefore 2 X 2 X 3 X 3: 36 is the least common multiple required. = = = SECOND OPERATION. 316 9 12 22 3 4 1 3 2 3 X 2 X 3 X 2 Having arranged the numbers on a horizontal line, we divide by 3, a prime number that will divide all of them without a remainder, and write the quotients in a line below. We next divide by 2, a prime number, writing down the quotients and undivided numbers as before. Then, since these numbers are prime to each other, we multiply together the divisors and the nun.bers on the lower line, which are all the prime factors of 6, 9, and 12, and thus obtain 36 for the least common multiple. 36 - 36 RULE 1. -Resolve the given numbers into their prime factors. The product of these factors, taking each factor the greatest number of times it occurs in any of the numbers, will be the least common multiple. Or, = RULE 2. Having arranged the numbers on a horizontal line, divide by such a prime number as will divide most of them without a remainder, and write the quotients and undivided numbers in a line beneath. So continue to divide until no prime number greater than 1 will divide two or more of them. The product of the divisors and the numbers of the line below will be the least common multiple. NOTE 1. When numbers are prime to each other, their product is their least common multiple. NOTE 2. -When any of the given numbers is a factor of any of the others it may be canceled. 128. What are the rules for finding the least common multiple ? EXAMPLES FOR PRACTICE. 2. What is the least common multiple of 7, 14, 21, and 15? Ans. 210. OPERATION. 77 14 21 15 3 15 2 7 X 2 X 15 210 3. What is the least common multiple of 3, 4, 5, 6, 7, and 8? Ans. 840. = Since 7 is a factor of 14, another of the numbers, we cancel it; and since 3 is a factor of 15, we also cancel that (Note 2); thus the work is rendered shorter. 4. What is the least number that 10, 12, 16, 20, and 24 will divide without a remainder? Ans. 240. 5. What is the least common multiple of 9, 8, 12, 18, 24, 36, and 72? Ans. 72. 6. Five men start from the same place to go round a certain island. The first can go round it in 10 days; the second, in 12 days; the third, in 16 days; the fourth, in 18 days; the fifth in 20 days. In what time will they all meet at the place from which they started? Ans. 720 days. FRACTIONS. 129. A Fraction is an expression denoting one or more equal parts of a unit. Fraction is derived from the Latin frango, to break. COMMON FRACTIONS. 130. A Common Fraction is expressed by two numbers, one written over the other, with a line between them. The number below the line is called the denominator; and the number above, the numerator. Thus, Numerator, 3, Three Denominator, 5, Fifths. 129. What is a fraction? From what is the term derived, and what does it signify? How many kinds of fractions, and what are they called? 130. How is a common fraction expressed? What is the number below the line called? The number above the line? The Denominator shows into how many parts the whole number is divided, and gives a name to the fraction. The Numerator shows how many of these parts are taken, or expressed by the fraction. A Proper Fraction is one whose numerator is less than the denominator; as, §. An Improper Fraction is one whose numerator is equal to, or greater than, the denominator; as, §, %. A Mixed Number is a whole number with a fraction; as, 761, 58. A Simple or Single Fraction has but one numerator and one denominator, and may be either proper or improper; as, 7, §. A Compound Fraction is a fraction of a fraction, connected by the word of; as, of % of §. A Complex Fraction is a fraction having a fraction or a mixed number for its numerator or denominator, or both; as, 2 7 81 71 7'97' 11' 9 131. The Terms of a fraction are its numerator and denom、 inator. The Unit of a Fraction is the unit or whole thing divided. A Fractional Unit is one of the equal parts into which the unit of the fraction is divided. A whole number may be expressed fractionally, by writing 1 for the denominator. Thus, 5 may be written, and read 5 ones; and 9 may be written, and read 9 ones. 132. Fractions originate from division; the numerator answers to the dividend, and the denominator to the divisor. Thus, when we divided 479956 by 6 (Art. 49, Ex. 12), we had a remainder of 4, which could not be divided by 6, and therefore we wrote it over the divisor, with a line between them. This expression originating from division is a fraction; the number above the line being the numerator, and the one below the denominator. 130 What does the denominator of a fraction show? What does the num rator show? What is a proper fraction? An improper fraction? A mixed number? A simple fraction? A compound fraction? A complex fraction?-131 What are the terms of a fraction? What is the unit of a fraction? How may a whole number be expressed fractionally? From what do fractions originate? |