(a'—b') x 3 a'= (a+b) 3ax (a+b) (a-b) 5bx (a+b) 3ax (a - b) is (dividing the numerator and denominator by a+b) 5b = (3x2-5x) × 7a=21ax2-35ax the numerator and denomi Зах — 5 а . nator by 7x) is the 4x2-6 6x3 ANSWER, 7x-7 On the Method of finding the Greatest Common Measure and the Least Common Multiple of two or more Quantities. 40. One quantity is said to measure another, when it is contained in that other a certain number of times, without a remainder. 41. A quantity is said to be a multiple of another, when it contains that other quantity a certain number of times, without a remainder. 42. The common measure of two or more quantities is that quantity which measures them all; and the greatest common measure is the greatest quantity which will so measure them. Thus, 2a is a common measure of the quantities 24 ab, 16a2bc, and 12abc, and their greatest common measure is 4 ab. 43. A common multiple of two or more quantities is that quantity in which each of them is contained without a remainder; and the least common multiple is the least quantity in which they are so contained. Thus, 40a3b'c is a common multiple of 5 a, 4 ac, and 2b, and their least common multiple is 20 abc. 44. If one quantity measures another, it will also measure any multiple of that quantity. Thus, let U measure a by the units in m, then amb; and let na be a multiple (denoted by the units in ) of a, then na-nmb; consequently b measures na by the units in nm. 45. If one quantity measures two others, it will also measure their sum and difference. For let c measure a by the units in m, and b by the units in n, then a=mc, and b=nc; therefore a+b=mc±nc,=(m±n)c; consequently c measures a+b (their sum) by the units in m+n, and a−b (their difference) by the units in m—n. 46. Let a and b be any two numbers, whereof a is the greatest; and let the following operation be performed upon them; viz. b) a (p pb c) b (q gc Where a divided by b gives the quotient p, and remainder c; b divided by c, the quotient, and remainder d; c divided by d, the quotient r, and remainder 0. Then, since in each case the dividend is equal to the divisor multiplied by the quotient plus the remainder, we have, b=qc+d= (for qc=qrd) qrd+d=(gr+1)d q=pb+c=(for pb=(pqr+p)d)(pgr+p+r)d. Hence, since p, q, r are whole numbers, d is contained in b as many times as there are units in qr+1, and in a as many times as there are units in pqr+p+r; consequently the last divisor d is a common measure of a and b; and this is evidently the case, whatever be the length of the operation, provided that it be carried on till the remainder is nothing. This last divisor d is also the greatest common measure of a and b. For let x be a common measure of a and b, such that a=mx, and b=nx, then c=a-pb=mx-pnx=(m-pn)x d=b-qc=nx (qm-pqn)x=(n−qm+pqn)x ; .. x mea sures (4) The quantity ab means a + or - b. sures d by the units in n-qm+pqn, and as it also measures a and b, the numbers a, b, and d have a common measure. Now the greatest common measure of d is itself; consequently dis the greatest common measure of a and b. Hence this Rule for finding the greatest common measure of two numbers; "Divide the greater by the lesser, and the preceding divisor by "the last remainder, till nothing remains; the last divisor is "the greatest common measure.” To find the greatest common measure of three numbers, a, b, c; let d be the greatest common measure of a and b, and x the greatest common measure of d and c; then x is the greatest common measure of a, b, and c. For, as a, b, and d, have a common measure; if d and c have also a cominon measure, that same number will measure a, b, and c; and if x be the greatest common measure of d and c, it will also be the greatest common measure of a, b, and c. In general, let there be any set of numbers, a, b, c, d, e, &c. ; and let x be the greatest common measure of a and b; y the greatest common measure of x and c; ≈ the greatest common measure of y and d; &c. &c.; then will y be the greatest common measure of a, b, c ; z the greatest common measure of a, b, c, d; &c. &c. 47. To find the greatest common measure of Algebraic quantities, the Rule, with respect to simple quantities, is, "to "find the greatest common measure of their coefficients, and "then annex to it the letters common to all the quantities;" thus the greatest common measure of 24ax'y', 16bxy, and 6axy'; is 2xy. The operation for finding the greatest common measure of compound algebraic/ quantities is the same as that for finding the greatest common measure of two numbers, except that "the remainders which arise in the operation are to be divided "by their simple divisors." In commencing the operation, the terms |