Divide 1749 by 477, and the remainder is 318. We infer that 477 is not the greatest measure; but (by § 112), any number which measures 1749 and 477 measures also 318, or the greatest common measure must be a factor of the three numbers, 1749, 477, and 318. We are thus guided in finding the greatest common measure of 477 and 318. Divide 477 by 318, and the remainder is 159; in the same manner, it is shown that the greatest common measure of 477, and 318 is a factor of the three numbers, 477, 318, and 159, Let us now ascertain the greatest common measure of 318 and 159. Divide 318 by 159, and the remainder is 0. Therefore, 159 is the greatest common measure of 318 and 159; but the greatest common measure of these two numbers is a measure of 477, 318, and 159; and, therefore, also of the four numbers 1749, 477, 318, and 159, or, in short, of 1749 and 477. Thus 159 is the greatest common measure required. The operation is effected in the following manner :— 477)1749(3 318)477(1 159)318(2 114. Therefore, to find the greatest common measure of two numbers, divide the greater number by the less; if there is a remainder, divide the less by it; continue dividing the last divisor by the last remainder till there is nothing remaining, and the last divisor is the common measure required. 115. If one of the given numbers were a prime number, there would be no common measure, unless the less number is the measure of the greater; of course, in this case, the less number is the prime number. 116. Strictly speaking, all numbers have a common measure, because they are all measured by 1; but when two numbers have no common measure greater than 1, or when they are prime to each other, the last divisor will be 1. If, in the operation, any divisor be a prime number, and leave a remainder, it is useless to proceed further, as unity is the only common measure. 117. If it were required to find the greatest common measure of three or more given numbers, we should ascertain the greatest common measure of any two of them; then that of this greatest common measure and a third number, and so on, to the end. EXERCISES. 8 6 72 1. Reduce to their lowest terms: 12, 15, 27, 73, 360. 30 2. Simplify, as much as possible, the following fractions: 38, 5565 592 345, $885, 888315,840 625 1440 8533' 999 3. Express, in their simplest forms, the fractions 59,9 1386 296T 94380 to their most 43365 8280 simple expressions. (The four following Articles may be omitted until pupils have acquired a better knowledge of fractions.) 118. We have established this principle: that fractions are not altered in value when both numerator and denominator are divided by the same number. But it often happens that fractions, even when reduced to their lowest terms, are expressed in numbers inconveniently high, and it is sometimes required to find approximate values of them in smaller numbers. : [Note signifies greater than, as 20>6; < signifies less than, as 6<20; .. signifies therefore; : signifies since or because.] Suppose we want to find an approximate value of 353, which is expressed in its lowest terms. Let us divide the terms of the fraction by the numerator, then the fraction becomes Were we to omit the fraction 154, the de 1 1 1213 3+ 353 154 Now, if we divide the fraction of the denominator by 154, we Proceeding with the fraction of the denominator as before, we is between It follows that the value of the given fraction is between and 2. To continue the approximation, let us divide both terms of 7 19 neglecting the fraction we 2+ By continuing the process in a similar manner, we find other values which approximate more and more towards 3513, until at last we obtain the given fraction. If the remaining fraction after each division be omitted, except when the numerator is unity, we have the following converging fractions :— 119. Ex. Find the fractions converging to 135. The following process is convenient to determine the several fractions approximating to 13 |