138 CONTINUED FRACTIONS. greatest number that can be a measure of them both; for any number which is so must be a measure of 18, and it is evident that 18 is the greatest number that can be so. Generalising this operation, we have the common rule for finding the greatest common measure of any two numbers :-divide the greater by the less, and the divisor by the remainder continually till nothing remains, and the last divisor is the greatest common measure. To reduce any fraction, or any ratio to its lowest terms, we have only to find the greatest common measure and divide both terms by it: thus, if our example were the ratio 6354: 8172, or the fraction 6354 454 the lowest term would be 353:454, or the fraction ; and if they are tried by the same operation it will be found that these numbers have no common divisor. Numbers which have no common divisor are said to be prime to each other; and such numbers taken singly may be either prime or composite. Any number of which several other numbers are factors or divisors is called a common multiple of them, and the least number of which they all are divisors is their least common multiple. But, before we examine the multiples of numbers, it will be of use to revert to the process by which we find the greatest common divisor, because that process is useful in practice, even though the result of it should be that the numbers have no common divisor. If we express the first step of the division in the former 1 example we have 8127 × 6354= ; if we substitute the 1188 6354 second division for the fraction in this denominator, we have 1 3900 1818 we have 1 50 ; if we substitute the fraction for the division, in this 1 218 900 ; and from this, by the last division, we have But the simple fraction in each of these belongs to the denominator of the fraction before it; therefore, the whole assumes this form : 1 11 31 21 50. This is called a continued fraction, because every following fraction is part of the denominator of the one before it. We may take the whole of this fraction and reduce it, which 6354 will give us the lowest terms of the fraction that is, 8172' 353. 1 The last two parts are 454 21; and multiplying both 250 terms of this by 50, to clear it of the fraction in the denomi nator, we have 1 x 50 2x50+1' -, which, performing the multiplica 50 tions and the addition, gives 101 Substitute this in the term above, and it becomes ; reduce this and there is 1x353+101 454 1 101 353 ; and reducing the same as we obtained by the 140 CONTINUED FRACTIONS. direct division of the terms by their greatest common measure. But we might not have occasion to make use of all the nicety of the lowest terms of the fraction, or the ratio (for it is the same in either case), and then the continued fraction furnishes us with a series of approximations by taking one, two, three, or more terms, at pleasure, from the beginning. 1 In the above fraction, the first term gives us = 454, which is by much too high; the first and second terms, give us 353 454' 353 454 3 4 The first, second, and third terms, 1 , as before. So that we have the series 137 1' 4' 9' each nearer the truth than the one before it, till we come to the last, which takes in all the terms, and is, in consequence, exactly true. But if we examine the way in which these terms are obtained, we find that the second is the first multiplied by the second quotient, with 1 added to the product of the denominator; and that each succeeding one is the one before it multiplied by the next quotient, and the one before that added to the product, as in this operation : If we compare the statements we can see that the difference of the values of every two adjoining ones is 1 divided by the 1 1 product of the denominators, thus is - greater than 1 7 7 1 less than and - is 36 9' 9 4086 1 4 greater than 353 33 is 44 Thus, if we take the smaller of the given numbers as the numerator of the fraction, or as the term compared with the standard in the ratio, the first fraction gives the value too high, the second too low, the third too high, and so on alternately, till we come to the truth in those cases which terminate, or without limit in those which do not. Each is thus nearer the truth than the difference between it and the next. Let us compare these by reducing them to a common denominator, that is, by multiplying the terms of each by all the denominators except its own, thus : 1 4084 7 9 is very near the truth, being only of 1 from it, which is so little, that for any common purpose 7 and 9 would do just as well as 353 and 454. We shall have occasion to take some further notice of continued fractions, on account of the assistance they give us in matters much more difficult than the present. 142 LEAST COMMON MULTIPLE. If it is borne in mind that the least common multiple of any number of numbers is the smallest number that can be divided by each of them without remainder, and that when they are all prime to each other this number is the product of them all, the following practical directions will be understood without any explanation:--write the numbers in a line after each other, divide them by their common factors or measures till no number can be found that will divide two of them; then multiply all the divisors and undivided numbers, and the product will be the least common multiple. Let it be required to find the least common multiple of 18, 24, 16, 15, 14, and 9. Arrange them 18, 24, 16, 15, 14, 9; 2 divides them all but 15 and 9, and the results are, 9, 12, 8, 15, 7, 9 × 2. 3 divides them all except 7 and 8, and the results are, 3, 4, 8, 5, 7, 3×2×3; 3 divides 3 and 3, and the results are, 1, 4, 8, 5, 7, 1 x 2 x 3 x 3. 4 divides 4 and 8, and the results, leaving out the 1s, which make nothing as multipliers, are, 2, 5, 7 x 2 x 3 x 3 x 4, which are not only prime to each other, but all prime numbers except the divisors; therefore, 2×5×7×2×3×4 = least common multiple. In such cases we can often get the product with very little trouble. In the above 2×5×2×4 = 80, 80 × 7 = 560, 560×9(3x3) = 5040. The continual product is a much larger number, being |