Ex. 3. How many times does £263. 8s. 114d. contain £37. 12s. 84d.? Here both numbers must be reduced to the lowest denomination, which is farthings. The divisor £37. 12s. 84d. is equivalent to 36130 farthings, and the dividend 36130)252910(7 £263. 8s. 114d. is equivalent to 252910 farthings; 252910 then dividing the latter by the former, we get for a quotient the abstract number 7; hence the former sum contains the latter 7 times. 41. It is worthy of remark, that when a concrete number is divided by an abstract number, the quotient is a concrete number of the same kind as the dividend; but when one concrete number is divided by another, the quotient is an abstract number. Thus 24 days divided by 6 gives 4 days for quotient, while 24 days divided by 6 days give the abstract number 4 for quotient. GREATEST COMMON MEASURE. 42. When one number divides another without remainder, or is contained an exact number of times in it, the former is said to measure the latter. Thus 4 is a measure of 28, but it is not a measure of 29. When one number is a measure of two others, it is called a common measure of these others; and the greatest of all the common measures of two numbers is called their greatest common measure. Thus 2, 3, 6 are common measures of 18 and 30, but 6 is their greatest common measure. 43. If one number measures two others, it measures their sum and difference. Thus 9 measures 18 and 45, and hence it must necessarily measure both 45+ 18 and 45 - 18, or 63 and 27. 44. If one number measures a second, it measures every number which the second measures. Thus 9 measures 18; and 18 measures 36, 54, 72, etc., all of which are evidently measured by 9. 45. Every number which measures both the dividend and divisor, measures the remainder also; and every common measure of the divisor and remainder is also a common measure of the dividend and divisor. Dividing 168 by 63, we get 2 for a quotient, and the remainder is 42; that is, 168 63 × 2 + 42, and therefore 42 is the difference between 168 and twice 63, or 42 168 63 x 2. Take now any number which measures both 168 and 63 as 3; then, since 3 measures 63, it measures 63+ 63, or 63 × 2 by Art. 44, hence (43) it measures 168 = = 63 x 2, and therefore it measures the remainder 42. The same holds for all other measures of 168 and 63; hence it follows, that every common measure of a divisor and dividend is also a common measure of the divisor and remainder. Again, since 168 = 63 × 2 + 42, let us take any common measure of the divisor 63, and the remainder 42 as 7; then 7 measures 63 x 2 by Art. 44, and hence (43) it measures 63 x 242, that is, it measures the dividend 168. From this it follows that there is no common measure of the remainder and divisor which is not also a common measure of the divisor and dividend. Hence the greatest common measure of the remainder and divisor is also the greatest common measure of the divisor and dividend; that is, 126 42)63(1 the greatest common measure of 42 and 63 is also 63)168(2 42 21)42(2 42 63)168(3 This process may sometimes be shortened by taking the quotient figure, so that when the divisor is multiplied by it, the product shall be greater than the dividend; because it is only the difference between 168 and 126 which we have to deal with in the preceding reasoning. Thus taking the first quotient figure 3 instead of 2, the product 189 differs from 168 only by 21; whereas, in the former division, the difference or remainder is 42. The last divisor, 21, is hence the greatest common measure of 63 and 168. Otherwise. 21)63(3 63 Since 63 = 7 x 9 = 7 x 3 x 3, and 168 = 7 × 24 therefore 7 x 3 21, is the greatest common measure = Hence, to find the greatest common measure of two numbers, divide the greater by the less, and then the divisor by the remainder; repeat this operation till an exact divisor is obtained; this will be the greatest common measure sought. 46. A prime number is one which can only be measured by unity, and a composite number is one which can be measured by some number greater than unity, or it is the product of two or more numbers. Also, two numbers are prime to each other when they have no common measure greater than unity. 47. To obtain the greatest common measure of three numbers, as 63, 168, and 189, we must first find that of 63 and 168, which is 21, and since it is manifest that the number sought is either 21 or some measure of it, we have only to find the greatest common measure of 21 and 189, which is 21, because it is an exact divisor of 189. This process is applicable to four or more numbers. LEAST COMMON MULTIPLE. 48. A multiple of any number is one which contains it an exact number of times; a common multiple of two or more numbers is one which contains each of them an exact number of times; and the least common multiple of two or more numbers is the least number which contains each of them an exact number of times. Thus 15 is a multiple of 3 or 5; 24 is a common multiple of 3 and 4; and 12 is their least common multiple. 49. One method of finding the least common multiple of two numbers is to divide their product by their greatest common measure. For take any two numbers, as 72 and 30, and resolve them into their prime factors; then, since 72 2 × 2 × 2 × 3 × 3, and 30 2 × 3 × 5, we see at once that their greatest common measure is 2 × 3 or 6, and that if (2 × 2 × 2 × 3 × 3) × (2 × 3 × 5) be divided by 2 × 3 or 6, the quotient will be equal to 2 × 2 × 3 × 2 × 3 × 5. This = product must be the least common multiple of 72 and 30, since no smaller number is exactly divisible by 2 × 2 × 2 × 3 × 3 and 2 × 3 × 5, or by 72 and 30; hence 2 × 2 × 3 × 2 × 3 × 5 = 360 = least common multiple, and 360 2 x 2 x 2 x 3 x 3 x 2 x 3 x 5 2 x 3 72 × 30 = 72 × 5 = 30 x 12 == 6 50. Hence it follows, that if two numbers be divided by all the numbers which will divide them exactly, the products of the divisors and quotients will be the least common multiple. Thus, to find the least common multiple of 576 and 312, we first divide by 12, and then the quotients by 2, and consequently 12 x 2 x 24 × 13 7488 the least common multiple required. It is also evident that 12 × 2, or 24, is the greatest common measure of 576 and 312. = = 12576 312 2 48 26 24 13 51. From these examples we perceive that the product of two numbers is a common multiple of each, and that if two numbers have a common measure, they also have a common multiple less than their product. Also, when the greatest common measure is unity, the least common multiple of the two numbers is their product. The rule then is :-to find the least common multiple of two numbers, find their greatest common measure, and divide their product by it; or divide either of the numbers by their greatest common measure, and multiply the quotient by the other. 52. The least common multiple of three or more numbers may be found very simply by the following process. Write the numbers in a line, and divide by any number that will divide them all, or two or more of them, and set down the quotients and undivided numbers (if any) in a line below. Divide the numbers in the second line in a similar manner, and continue the operation until every two of the numbers are prime to each other. Then the continued product of all the divisors, and the numbers in the last line, will be the least common multiple of all the numbers. 2126, 168, 210, 294 3 63, 84, 105, 147 7 21, 28, 35, 49 Ex. 1. Find the least common multiple of 126, 168, 210, and 294. Beginning with the least prime divisor, 2, we find the quotients to be as in the second line. As 84 is the only number in this line divisible by 2, we try 3, the next prime number, and thus the third line is obtained; and as no two of these numbers are divisible by 2, 3, or 5, we take the next prime number 7, and this gives the quotients in the last line, every two of which are prime to each other; hence, 2 x 3 x 7 x 3 x 4 x 5 x 7 = 17640 least common multiple sought. = the The greatest common measure of all these numbers is evidently the product of all the divisors 2, 3, 7, or 42; and this being the greatest common factor of all the numbers, it is evident that if the product of all the other factors, 3, 4, 5, 7, be multiplied by this number 42, the least common multiple of all will be obtained. Ex. 2. Find the least common multiple of all the nine digits. Here we may either take all the digits, or omit 2, 3, 4; because, whatever number is measured by 9 and 8, will be measured by 2, 3, 4. In either way we have the least common multiple = 2 × 3 × 5 x 7 x 4 × 3 = 2520. 53. The least common multiple of several numbers may be found by first finding the least common multiple of any two of them; then the least common multiple of that multiple, and a third number; and 54. The nature of a fraction will be understood by supposing that a unit of any kind is divided into several equal parts. One or more of these parts is called a fraction of that unit, and is represented by one number above a line, and another under it. The number under the line denotes the number of parts into which the unit is divided, and is called the denominator of the fraction, and the number above the line shows how many of these parts are represented, and is called the numerator of 3 8 the fraction. Thus if the unit be a yard, the fraction signifies that the yard is divided into 8 equal parts, and three of these parts are taken. It is read three-eighths. If the unit be divided into 2, 3, 4, 5, or 6 equal parts, the corresponding fractions, with their names, are as follow: 1 2' 1 3' 1 4' 1 1 6' one-half; }, one-third; }, one-fourth; }, one-fifth; one-sixth. 55. Hence it follows that if the numerator of a fraction be less, equal to, or greater than the denominator, its value is less, equal to, or greater than unity. 56. A proper fraction is one whose numerator is less than the denominator; and an improper fraction is one whose numerator is either equal to, or greater than the denominator. Thus 3 9 5' 10 are proper frac In the latter case we may suppose each of two or more units to be divided into the number of parts indicated by the denominator, and from these units so divided, take as many parts as there are units in the numerator. 57. A simple fraction has only one numerator and one denominator, which are called the terms of the fraction, as 3 2 9 and A compound 4' 5' 8 fraction is two or more fractions with the word of between them; thus is a compound fraction. A mixed number is composed of an integer and a fraction, as 1, 6, and 154; and a complex fraction has a fraction or a mixed number in one or both of its terms, as 12 5' 58. From the notion attached to the words numerator and denominator, it is evident that 1. A fraction is increased by increasing the numerator. For by increasing or diminishing the numerator, we take more or fewer of the parts of the unit; by diminishing the denominator, the magnitude of the parts is increased, while the same number of parts are taken. Also if the denominator be increased, the magnitude of the parts is diminished, and the fraction is diminished. Thus the fraction 59. Hence a fraction is multiplied or divided by multiplying or dividing the numerator; and a fraction is divided or multiplied by multiplying or dividing the denominator. Also if the numerator and denominator of a fraction be multiplied or divided by the same number, the value of the fraction is not altered. It is worthy of remark that, by suppressing the denominator of a fraction, it becomes multiplied by this number. pressing the denominator 8 in the fraction For example, by sup it becomes 3 whole num 3 8' 3 bers, or is multiplied by 8; that is x8 = 8 3. Also since the integer 3 8' 3 is 8 times greater than the fraction it is evident that 3 may be ex pressed in a fractional form by writing 1 in the denominator; thus 3 and so on. 60. To reduce a fraction to its lowest terms. By Art. 59, the value of a fraction is not altered if both numerator and denominator be divided by the same number; therefore divide the |
