the number itself. Thus 24, which is equal to 2×2× 2 × 3, contains 2×2 or 4, 2× 3 or 6, 2×2×2 or 8, and 2×2× 3 or 12. Then the least common multiple of any series of numbers, is the least number which contains all the factors of the given numbers, and may be found by the following RULE. Arrange the numbers in a horizontal line, and divide successively by the prime numbers 2, 3, 5, 7, 11, &c., employing each divisor as often as it will divide one or more of the numbers without a remainder, writing the quotients and undivided numbers, beneath. Continue this division until the last quotients are all 1, and you will have obtained all the prime factors of the given numbers. The product of these factors, is the least common multiple. By the table of prime factors, the least common multiple can be found much more readily, in the following manner. Form the product of all the prime factors of the given numbers, employing each factor the largest number of times it is used in either number. Referring to the numbers in the foregoing example, we find they are respectively equal to 2×7, 2×3×3, 3×3×3, 3×7, 2X2X7, and 2×3×3×7. The only prime numbers used, are 2, 3, and 7. 2 is employed 2 times in the 5th number, 3 is employed 3 times in the 3d number, and 7 is employed but once in either number. The least common multiple is then, 2×2×3× 3x3x7=756, as before. 1. Find the least common multiple of 96, 32, 12, 24, and 48.4 2. Find the least common multiple of 32, 27, 16, 18, and 54. 3. Find the least common multiple of 35, 20, 28, 21, and 12. 20 4. Find the least common multiple of 19, 27, 18, 12, and 36. 5. Find the least common multiple of 2, 3, 4, 5, 11, 12, and 15. 6. Find the least common multiple of 21, 25, 12, 18, and 49. 7. Find the least common multiple of 9, 11, 14, 63, and 99. 8. Find the least common multiple of 10, 6, 42, 15, 30, 105, and 210.216 Any number that will exactly divide two or more other numbers, is called a common divisor, or common measure, and the greatest number that will so divide them, is the greatest common divisor, or greatest common measure, of those numbers. Thus 2, 3, and 6, are all common divisors of 6, 12, 18, 24, and 30; but their greatest common divisor is 6. Any two numbers that have no common divisor, as 6 and 7, 11 and 15, 18 and 25, are said to be prime to each other. EXAMPLE FOR THE BOARD. The common divisor of any two numbers, will also divide their difference. Thus 3 is contained in 27, 9 times, and in 36, 12 times. It must therefore be contained in 36-27, 12-9, or 3 times. We Let it then be required to find the greatest common divisor of 384 and 672. first divide 672 by 384, to see if the smaller number will exactly divide the larger, and we find a remainder, 288. Now the greatest common divisor of 384 and 672, is also the greatest common divisor of 288 and 384, because, if we suppose it contained 384)672(1 288)384(1 96)288(3 288 4 times in 384, and 7 times in 672, it must be contained 3 times in their difference, 288. Dividing 384 by 288, we find a remain der, 96. For the reason given above, the greatest common divisor sought, is also the greatest common divisor of 96 and 288. Dividing 288 by 96, we find it is contained exactly 3 times. 96 is therefore the greatest common divisor. Hence the following RULE. Divide the larger number by the smaller, and if there is no remainder, the smaller number will be the greatest common divisor. If there is a remainder, divide the first divisor by the first remainder, the second divisor by the second remainder, and so proceed until you obtain a quotient without a remainder. The last divisor will be the greatest common measure. If the divisor of more than two numbers is required, first find the common divisor of any two, then of this divisor and a third, and so on. By the table of prime factors, the greatest common divisor may be found more readily in the following manner: Form the product of the prime factors common to all the given numbers, employing each factor, the least number of times it is used in either number. What is the greatest common divisor of 430, 602, 2150, and 3612? By the table we find these numbers are equal, respectively, to 2×5×43, 2×7×43, 2×5×5×43, 2×2×3×7x43. The only factors common to all, are 2× 43=86, which is the greatest common divisor. 1. Find the greatest common divisor of 48, 72, and 60. ' 2. Find the greatest common divisor of 1001, 385, and 539. 3. Find the greatest common divisor of 405, 567, 729, and 891. 4. Find the greatest common divisor of 2863 and 1151. / 5. Find the greatest common divisor of 992, 960, 928, and 32. 6. Find the greatest common divisor of 1177, 1391, and 1819. 7. Find the greatest common divisor of 2943, 2616, and 4578. 8. Find the greatest common divisor of 2148, 6444, 3580. and CHAPTER VI. FRACTIONS. If we divide 16 apples among 3 boys, we can give each of them 5, and have 1 left. If we wish to divide the remaining apple among them, we must cut it into 3 equal parts, and give one to each of them. Each of those parts would be called one-third, and written . Again, if we wished to divide 31 apples among 9 boys, we could give each of them 3, and have 4 left. To divide those 4, we might cut each one into 9 parts, or ninths, and give each boy 4 of the parts. Then 4÷9=3. Numbers of this kind are called Fractions, or broken numbers, and may be regarded in three different lights. Thus, may be read 3-fourths, of 3, or 3 divided by 4; is 31 fifty-sixths, of 31, or 31-56, &c. 56 The dividend, or numerator, may be considered as numbering the parts that are taken. The divisor, or denominator, marks the number of parts into which a unit is divided. The numerator and denominator are also called the terms of the fraction. A proper fraction, is one in which the numerator is less than the denominator, and the fraction is therefore less than 1, as, 27. 22 An improper fraction, is one in which the numerator is equal to, or greater than the denominator. In the former case, the fraction is equal to 1,-as 11, 3, 19. In the latter, it is more than 1, as 21, which (as =1) is equal to 3; 193; 178. A whole number may always be regarded as an improper fraction, whose denominator is 1. Thus, 12=2; 28=28. A mixed number, consists of a whole number and a fraction, as 33, which (as 1=3) is equal to 11; 12619, which (as 1=13) is equal to 1618. A compound fraction, is a fraction of a fraction, as of 2; 4 of 3 of 777· A complex fraction, is one which contains a fraction in its numerator or denominator, as ; 81 REDUCTION OF FRACTIONS. An improper fraction may be reduced to a whole or mixed number, by performing the division which the frac tion expresses. Thus, 27-27-11-25. This reduction must always be made in giving the result of any operation. A whole number may be reduced to a fraction having any given denominator, by multiplying it by the denominator. Thus, 2=22=23=8. A mixed number may be reduced to an improper fraction, by reducing the whole number to a fraction, and adding the fraction. Thus, since 2=2?,2,5,=??+5=}7. A compound fraction may be reduced to a simple one, by multiplying all the numerators together for a new numerator, and all the denominators for a new denominator. Thus, of. For of 1, because, if we suppose a unit divided into 6 equal parts, and each part again divided into 4 equal parts, the whole unit will be divided into 24 equal parts, or 24ths. Then, of =2, and of 14, and so of any other similar fractions. If the numerators and denominators have common factors, they may be rejected, as in the following example: 15 Reduce of of of 1, to a simple fraction. 1X2X5X14 2X7X9X15° Rejecting the common factors 2, 7 and 5, we have 1X2 9X3 The reason of this process will be evident, if we remember that 2X7X5 =1, and dividing any number by 1, does not alter its 2X7X5 value. Compound fractions must always be reduced to simple ones, before performing any operation. 1. Reduce to a whole or mixed number, 149; 270; 759; 421. 2. Reduce 49 to 7ths; 12ths; 13ths; 60ths. 3. Reduce to an improper fraction, 473; 26; 1493; 2273. 13 4. Reduce to a simple fraction, of of of; } of 5 of 12 of ; 4 of 1 of 3 of 3; 5. Reduce 19 of 11 of of to an improper fraction; 3 to a mixed |