Let it be required to find the least common multiple of 12, 15, 7, 18, 3, 5 and 35. 712, 15, 7, 18, 3, 5, 35, First, dividing by the prime number 7, I find that it divides two of the given numbers, 7 and 35. The rest being written in the line below, I take 5, for the next divisor, which divides 15 and 5. Continuing the operation in the same manner, I divide successively, by 3, 3 and 2, when the division can be carried no further; then multiplying all the divisors together, and that product by 2, (the only number greater than unity found in the lower line), the result is 1260. Now, to show that 1260 is a multiple of each of the given numbers, it is only necessary to observe, that among its prime factors, 7, 5, 3, 3, 2, 2, we can find three of the given numbers, namely, 7, 5 and 3; it must, therefore, be divisible by each of these. Moreover, the prime factors of each of the other numbers, are found among the factors of 1260, and, therefore, it is divisible by each of those numbers; for example, 3×2×2 = 12. And it is evident, that 3×2×2×5×7×3, or, 12 x 5 x 7 x 3 = 1260 is divisible by 12; and that 3x5x7×3×2×2, or, 15×7×3×2×2=1260 is divisible by 15, and so of the others. It is also the least common multiple of the given numbers; for it has, among its factors, those only, which are necessary to make up the prime factors of the given numbers. It is also evident that the rule will apply to any numbers whatever, since it is nothing more than finding the prime factors of the given numbers, and taking their product. 12. EXAMPLES. 1. Find the least common multiple of 9, 16, 35, 20 and 2×3×2×5×3×4x7=5040, the least common multiple. 2. Find the least common multiple of 3, 4, 9, 10, 21, 35 and 12. Ans. 1260. 3. Find the least common multiple of 2, 3, 4, 5, 6, 7, 8 and 9. Ans. 2520. 4. Find the least common multiple of 16, 11, 33, 27 and 45. Ans. 23,760. To reduce fractions to the least common denominator. RULE. (39.) Find the least common multiple of all the denominators; this will be the common denominator. For the numerators, divide the common denominator by each particular denominator, and multiply the quotient by the numerator. 7×2×8×3=336, least common denominator. 2. Reduce, and 4, to their least common denomi nator. 3 3. Reduce, 7, 4% and 71, to their least common deno minator. Ans. 150' 130' 150° 4 5 147 142 Τ 4. Reduce, 21, and 8, to their least common denomi 6. Reduce of, and to their least common denomi nator. Ans., 36. 560 4 ADDITION OF FRACTIONS. (40). Before fractions can be added together, they must be reduced to a common denominator, so that the parts of an unit, expressed by each fraction, may be the same (18). For example, the sum of and can neither be expressed in thirds nor fourths. But, since = 2, and 1 12, their sum can be expressed in twelfths, and will be equal to 7. Hence we have, for the addition of fractions, the following RULE. 4 Reduce the fractions to a common denominator, and write the sum of the numerators over the common denominator. EXAMPLES. 1. Find the sum of 2, and 3. These fractions reduced to a common denominator become, 17, 14 and 18. Taking the sum of the numerators, we have for the sum 67 of the fractions, 7 or, 131. 36 When mixed numbers are to be added, they may be reduced to improper fractions; or, the fractional parts may be added separately, and the whole numbers added to the sum. 5. Find the sum of 51, 35, 13 and 2. Here the sum of the fractional parts is 25; the sum of the whole numbers is 9; the sum of the whole is, therefore, 115. 6. Find the sum 31, 3, 6 and 28. 22 Ans. 3953. SUBTRACTION OF FRACTIONS. RULE. (41.) Prepare the fractions as in addition, and write the difference of the numerator over the common denomi nator. EXAMPLES. from 3. 159 1. Subtract 3 15 5 and = 5; 15-£= 1⁄2, the answer. 2. Subtract from 12 3. Subtract from 72. 4. Subtract of 4 from 53. 5. Subtract 3 from 53. 6. Subtract 1 from 5 Ans. T Ans. 43. Ans. 213. Ans. or 117. EXAMPLES IN ADDITION AND SUBTRACTION. 1. Find the sum of -+-}. Ans.. In examples like the preceding, all the fractions having the sign+, are to be collected into one sum, and all those having the sign, into another, and the difference between these sums taken. 2. Find the sum of --+ -}} Ans. 13 Ans. 1. Ans. 153. CONTINUED FRACTIONS. (42.) A continued fraction is one which has a whole number, and a fraction for its denominator, the fractional part of which, has also a whole number and fraction for its denominator, and so on. For example, 1 3+1 2+1 1+ is a continued fraction. A continued fraction is formed from a simple fraction, the terms of which are prime to each other, by dividing both terms by the numerator. The denominator not being exactly divisible by the numerator, will become a mixed number, the fractional part of which is divided as before. In the example above, the original fraction was. ding both terms of this fraction by 11, it becomes dividing both terms of by 4, we obtain 1 2+3/ Divi 1 311 again stituting this in the place of, the fraction becomes 1 3+1 2+¥ Operating in the same manner upon 2, we find which, being substituted in the place of in the continued fraction, it becomes |