THE LEAST COMMON MULTIPLE OF NUMBERS. A MULTIPLE of a number is any number that contains the other a certain number of times exactly: thus, 12 is a multiple of 3. A COMMON MULTIPLE of two or more numbers is any number that contains each of them so many times exactly: thus, 48 is a common multiple of 8 and 12. THE LEAST COMMON MULTIPLE of two or more numbers, is the least number that is a multiple of each of the given numbers thus, 24 is the least common multiple of 4, 6, 8, 12. : TO FIND RULE.-1. Write the given numbers in a line, one after the other; cancel such of them as divide any of the others exactly; then divide as many of the rest as practicable, by some number that divides them without a remainder, placing the quotients and any undivided numbers in the line below. * 2. Cancel, divide, &c., the numbers in this line as before, carrying on the process till no numbers remain that have a common measure. 3. Multiply together the numbers used as divisors, and any undivided numbers in the last line; and the product is the least common multiple required. This rule, and the preceding, is useful in the reduction of Fractions. Note. If no two of the given numbers have a common measure, their continued product will be the least common multiple. Example. Find the least common multiple of 16, 20, 4, 25, 12, and 18. 2)16, 20, 4, 25, 12, 18 4, 5, 25, 9, 9 Pupil. In this example the numbers are placed in a line, and as 4 is a factor of 16, it is rejected. Using 2 as a divisor, I place the quotients 8, 10, 6, 9, and the undivided number 25, in a new line. Again, using 2 as a divisor, I place the quotients 4, 5, 3, and the undivided numbers 25 and 9, in a new line. Now, as 5 is a factor of 25, and 3 a factor . 2 × 2 × 4 × 25 × 9 = 3600 Ans. * Take whatever prime number will divide more of them than any other that could be taken. of 9, the numbers 5 and 3 are rejected; and as no two of the remaining numbers, 4, 25, and 9, have a common measure greater than 1, therefore, the two divisors, 2, 2, and the undivided numbers, 4, 25, 9, multiplied together, give 3600, the least common multiple required. REASON OF THE RULE.-In the foregoing example, since 16 is a multiple of 4, therefore, any multiple of 16 is divisible by 4; hence, the least common multiple of 16, 20, 4, 25, 12, and 18, is also the least common multiple of 16, 20, 25, 12, and 18. Wherefore, in finding the least common multiple of the given numbers, the 4 is rejected. Again, the least common multiple must be divisible by all the factors into which each of the given numbers, 16, 20, 25, 12, and 18, can be resolved, since it is divisible by each of the numbers; it must therefore contain as factors all the factors of the given numbers, and no more. Hence, when any factors are common to any two or more of the given numbers, these factors are retained in one of the numbers, and removed at once, or gradually, from the others, which is what the rule prescribes. Exercises. 1. Required the least common multiple of 5, 18, 16, 2, 9, Ans. 720. 2. 1234 4. 5. 6. 7. 8. VULGAR FRACTIONS. A FRACTION means a part of a whole: the term is derived from a Latin word signifying broken. Whole or unbroken numbers, as 1, 2, 3, &c., are termed integers; broken numbers, as, 1, a half; }, a third, &c., are termed fractions. Fractions are of two kinds: vulgar fractions, from a Latin word signifying common; and decimals, from a word signifying ten. VULGAR FRACTIONS are the common fractions of halves, thirds, fourths, and so on; the term is applied to all fractions when expressed by figures in this form-, two-thirds; %, five-sixths, &c. They are called vulgar fractions, in distinction from decimal fractions (see page 135). If we suppose a loaf to be divided into two equal parts, each of the parts is a half, and forms a fraction of the whole; in figures, it is written as a Vulgar Fraction, thus-. Again, if the loaf is divided into four equal parts; each of these is called a fourth or a quarter, and is written thus-; two of them may be expressed as 4, but as two-fourths are the same as one-half, they are written, ; three of them are written, 3, expressing three-fourths. If the whole be divided into three equal parts, each part is called a third; if into five, each is called a fifth; if into six, a sixth; and so on, according to the number of parts into which the whole is divided; thus- means two-thirds of a whole; 3, three-fifths; g, five-sixths. To represent a Vulgar Fraction, therefore, two numbers are required, which are written the one above the other, with a short line between. The number under the line shews into how many parts the whole of the article, whatever it may be, is divided; and the number above the line shews how many of these parts we mean to express. The upper number is called the numerator, because it shews the number of the parts-as three-fourths, six-sevenths; the lower number is called the denominator, because it denominates the nature of the fraction-such as thirds, eighths, & All the parts are together equal to the whole. Thus-twohalves, or three-thirds, or four-quarters, make each a whole. The numerator and denominator are called the terms of the fraction. A fraction may therefore be considered as the result of two indicated operations; namely, division and multiplication. For example, indicates that the principal unit is first divided into three equal parts; and, secondly, that one of these parts is taken two times, or multiplied by 2, and in this view is considered as two-thirds of 1. Now, when multiplication and division are to be performed in succession, it is a matter of no consequence whether we first multiply and then divide, or first divide and then multiply. Hence, taking the number 1, and multiplying it by 2, we have 2; and indicating the division of 2 by 3, we have, which, in this view, is considered as the one-third of 2; hence the of 1 and the 3 of 2 are equal. Hence, a fraction may be viewed as indicating the division of the numerator by the denominator. From the nature of the notation of a fraction, a fraction is multiplied by any number, by multiplying the numerator of the fraction by that number; and a fraction is divided by any number, by multiplying the denominator by that number. The value of a fraction is not altered by multiplying or dividing both of its terms by the same number. Thus, is equal in 3 × 2 multiplied by 2 gives and this product 4 3 × 2 4 X 2 value to ; for divided by 2 gives ; now, when a number is first multiplied by 2, and then the product divided by 2, the value of the number is not altered,.. 3 x 2 = g. If a class of 4 boys have 3 oranges divided among them, and a second class of 8 boys have 6 oranges divided among them, it is obvious that one of the first class and one of the second would receive the same part of an orange. As the principles now stated are of the utmost importance, we shall establish their truth in another manner. Twice the third part of 1 is the same in value as the third part of 2, or of 1 = of 2.* Let AB be taken equal to 2 feet, for example, and divide each of the feet AC and CB into 3 equal parts. It is obvious that foot. Again, AE, EF, and FB are all equal to one another; therefore, AE is equal to the third part of 2 feet. Hence, to obtain the length, it makes no difference whether we first divide 1 foot into 3 equal parts, and then take 2 of them, or divide 2 feet into 3 equal parts, and take 1 of them. * De Morgan. The value of a fraction is not altered by multiplying or dividing both its terms by the same number: thus, = . Let AB represent a foot; divide it into 4 equal parts, AC, CD, DE, EB, and divide each of -B these into 2 equal parts. Then AE is; but the second division divides AB into 8 equal parts, of which AE contains 6; it is therefore §. Hence. That is obvious. When the numerator of a fraction is equal to its denominator, the fraction is equal to 1. For example, the fractions §, 8, 11, are each equal to 1: this is very obvious. When the numerator of a fraction is less than its denominator, the fraction is less than 1. For example, is less than 1, for is is less than 1. 용 less than ; but & is equal to 1, therefore, When the numerator is greater than the denominator, the fraction is greater than 1. For example, is greater than 1, for is greater than ; but is equal to 1, therefore, is greater than 1. A PROPER FRACTION is one whose numerator is less than its denominator; as . AN IMPROPER FRACTION is one whose numerator is not less than its denominator; as,. NUMBERS that are not fractional are called INTEGERS, or WHOLE NUMBERS, to distinguish them from fractions. A MIXED NUMBER consists of a whole number and a fraction, and is expressed by writing the whole number before the fraction; as 51. Since a fraction may be divided into a number of equal parts, and any number of these parts be taken, we obtain an idea of the fraction of a fraction. Two-thirds of five-eighths, written of, means that the fraction is to be divided into three equal parts, and that two of them are taken. Such a fraction is termed a compound fraction. A COMPOUND FRACTION, then, is the fraction of a fraction, or any number of fractions connected by the word of; as of of A COMPLEX FRACTION is one which has a fraction or mixed number for its numerator, or denominator, or both; as 32 6 54 7' 4' 93 Note. An integer may be reduced to a fractional form by putting 1 for its denominator; thus, 3, when changed to a fractional form, becomes . One whole number may be considered as the fraction of another; thus, we say 3 is the fourth of 12, that 5 is the five-sevenths of 7, &c. |