and so on, until the smaller paper would have been exhausted— there would remain, of the larger paper, the difference between what it contained and what the smaller paper contained; and this difference would be in the form of one or more rows of 12 each. General demonstration.-Let x and y be two numbers, of which x is the greater, and let both be measured by m: to prove that x+y and x-y are also measured by m. Let x and y be divided by m, and let the resulting quotients (there can be no remainders) be x' and y', respectively. We then have m)x x' y y' x=mx'; y=my'; x+y=mx'+my'=m(x'+y'); x—y=mx'—my'=m(x'—y'); x=mx', and y=my' ; x+y= mx'+my'=m(x'+y'), and (x+y)÷m=x'+y' ; x—y=mx'—my'—m(x'—y'); (x-y)÷m=x'y'. (x+y)÷m=m(x'+y')÷ It thus m=x'+y', and (x—y) ÷m=m (x'—y')÷m=x'—y'. appears that m is contained an exact number of times in x+y, and also in x-y: in other words, that both x+y and x—y are measured by m. 92)438(4 70)92(1 22)70(3 Let us now return to Example I. :-Beginning at the bottom, we see that the "last divisor,”" 2, being a measure of 4, measures 20, a multiple of 4; that, measuring 20 and itself, 2 measures 22-the sum of 20 and itself; that, measuring 22, 2 measures 66—a multiple of 22; that, measuring 66 and 4 (the last dividend), 2 measures 70the sum of 66 and 4; that, measuring 70 and (as we have already seen) 22, 2 measures 92-the sum of 70 and 22; that, measuring 92, 2 measures 368-a multiple of 92; and that, measuring 368 and (as we have already seen) 70, 2 measures 438-the sum of 368 and 70. 4)22(5 20 -- 2)4(2 4 O So that 2 is, at all events, a common measure of 92 and 438. It is also the greatest common measure. For, if 92 and 438 were both measured by a larger number than 2, this larger number would measure (92 X4=) 368, and (438—368=) 70, and (92-70-) 22, and (22 × 3=) 66, and (70—66—) 4, and (4 × 5=) 20, and (22—20=) 2. But a number larger than 2 cannot measure 2: neither, therefore, can a number larger than 2 be a common measure of 92 and 438. So that 2 is not only a common measure, but the greatest common measure, of 92 and 438. General demonstration.-Let x and y be two numbers, of which y is the greater. Let the division of y by x give a for quotient and (y-ax=) b for remainder; the division of x by b, c for quotient and (x-bc=) d for remainder; the division of b by d, e for quotient and (b-de-) f for remainder; and the division of d by f, g for quotient, and no remainder to prove that ƒ is the greatest common measure of x and y. : x) y (a ах b) x (c d) be f) d (g fg The division of d by f leaving no remainder, f is a measure of d, and therefore of de, and (f measuring itself) of (de+f=) b, and of bc, and of (bc+d=) x, and of ax, and of (ax+b=) y. So that ƒ is a common measure of x and y. It is also the greatest common measure. For, if x and y were both measured by a larger number than f, this larger number would measure ax, and (y-ax=) b, and bc, and (x-bc=) d, and de, and (b-de=) f. But a larger number than f cannot measure f: neither, therefore, can a larger number than f be a common measure of x and y. Consequently f, which has been shown to be a common measure, is the greatest common measure of x and y. NOTE.-The greatest common measure of two numbers is measured by every other common measure. For we have just seen that any number which measures x and y must measure their greatest common measure, f; and in the case of Ex. I. we saw that no number could measure 92 and 438 without measuring their greatest common measure, 2. The common measures of 24 and 36, as we have already seen, are 1, 2, 3, 4, 6, and 12; and it will be observed that 12, the greatest common measure, is measured by each of the others. 97. To find the Greatest Common Measure of more than two numbers: Find the greatest common measure-first, of two of the numbers; next, of this common measure and a third number; then, of this new common measure and a fourth number; and so on, until all the numbers have been taken into account. The last common measure will be the greatest common measure required. EXAMPLE.-Find the greatest common measure of 72, 84, 102, and 135. The greatest common measure of 72 and 84 is 12; of 12 and 102, 6; and of 6 and 135, 3. So that, according to the Rule, 3 is the greatest common measure required : 72 84 102 135 Reason of the Rule.-Measuring 135 and 6, 3 measures 135, 102, and 12-the last two numbers being multiples of 6; and, measuring 12, 3 measures 72 and 84, which are multiples of 12. So that 3 is a common measure of 72, 84, 102, and 135. 12 درة It is also the greatest common measure, as we shall find if we remember that (Note, p. 109) "the greatest common measure of two numbers is measured by every other common measure." For, if a larger number than 3 were a common measure of 72, 84, 102, and 135, this larger number-measuring 72 and 84-would measure 12, the greatest common measure of 72 and 84; andmeasuring 12 and 102-would measure 6, the greatest common measure of 12 and 102; and, lastly, measuring 6 and 135,would measure 3, the greatest common measure of 6 and 135. But a larger number than 3 cannot measure 3: neither, therefore, can a larger number than 3 be a common measure of 72, 84, 102, and 135. So that 3 is not only a common measure, but the greatest common measure, of the given numbers. General demonstration.-Let w, x, y, and w z be four numbers, and let the greatest common measure of w and x be a; of u and y, b; and of b and z, c: to prove that c is the greatest common measure of w, x, y, and z. a x y Measuring z and b, c measures z, y, and a-the last two numbers being multiples of b; and, measuring z, y, and a, c measures z, y, x, and w—the last two numbers being multiples So that c is a common measure of the four given of a. numbers. It is also the greatest common measure. For, if a larger number than c were a common measure, this larger numbermeasuring w and x-would measure a, the greatest common measure of w and x; and—measuring a and y-would measure b, the greatest common measure of a and y; and, lastly,— measuring b and z,—would measure c, the greatest common measure of b and z. But a larger number than c cannot measure c: neither, therefore, can a larger number than c be a common measure of w, x, y, and z. Consequently c, which has been proved to be a common measure, is the greatest common measure of the given numbers. 98. When considered with reference to two or more of its measures, a number is called a COMMON multiple of those measures. Thus, we say that 24 is a "common" multiple of 4 and 6; that 36 is a "common" multiple of 4, 6, and 9; &c. 99. By the LEAST common multiple of two or more numbers is meant the smallest (or "least") multiple which those numbers have in common. Taking 12 and 8, for example, we see that the multiples of 12 are: 12 itself, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, &c.; that the multiples of 8 are: 8 itself, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, &c.; that the numbers which, although multiples of 12, are not multiples of 8, are: 12, 36, 60, 84, 108, 132, &c.; that the numbers which, although multiples of 8, are not multiples of 12, are: 8, 16, 32, 40, 56, 64, 80, 88, 104, 112, 128, 136, &c.; that the common multiples of 12 and 8 are: 24, 48, 72, 96, 120, 144, &c.; and that the LEAST common multiple is 24. Multiples of 12. 12 후후 468 120 432 144 &c. -Common multiples▬▬ 24 48 120 of 12 and 8. 8 72 &c. 96 104 Multiples of 8. 100. To find the Least Common Multiple of two numbers: Divide the product of the numbers by their greatest common measure.* EXAMPLE.-What is the least common multiple of 12 and 20? *This is most easily effected when we divide one of the numbers by the greatest common measure, and multiply the other number by the resulting quotient. The greatest common measure of the two numbers being 4, 4 -60. NOTE. The least common multiple of two numbers which are prime to one another is their product, the greatest common measure being unity. Thus, the least common multiple of 15 and 22— the greatest common measure being 1-is (15x2215 x 22. Reason of the Rule.-In examining the reason of the foregoing Rule, we must remember that— Every other common multiple of two numbers is a multiple of the least common multiple. Thus, the least common multiple of 12 and 8 is 24, and every other common multiple (48, 72, 96, 120, 144, &c.) is a multiple of 24. In like manner, the least common multiple of 15 and 10 is 30, and every other common multiple (60, 90, 120, 150, 180, &c.) is a multiple of 30. lq Passing from numbers to general symbols, let us put x and y to represent two numbers, 7 their least common multiple, and m any other common multiple. Now, if m be not a multiple of 7, the division of m by 7 will leave a remainder. Let / be contained 9 times in m, and let the subtraction of lg from m leave the remainder r, if possible. Then, x and y, being measures of l, are measures of lg-a multiple of 7; x and y are also measures of m; consequently x and y are measures of r-the difference between lq and m. In other words, ris a common multiple of x and y. But this is impossible, the least common multiple of x and y being 1, and r (the "remainder") being less than 7 (the "divisor"). So that the division of m by I can leave no remainder: in other words, m is a multiple of 7. Returning to the last example, and remembering that the product of two numbers is a common multiple of those numbers, we see that 12 x 20 is a common multiple of 12 and 20. Now, 12 x 20, if not the least common multiple, must be a multiple of the least, which, consequently, will be obtained from the division of 12 x 20 by some whole number. If we try 10 for divisor, we shall find that is a multiple of 12, but not of 20; 10 20 12 X 20. ΙΟ being a measure of 20, but not of 12. For, 10 being a measure of 20, 18 is equivalent to a whole number (2); which is the product of 12 by 28, is a multiple so that 12 X 20 of 12. But as 10 is not a measure of 12, 1 is not equivalent to a whole number; so that 12 X 20 ΙΟ which is the product of 20 |