A and B, and let m be prime to B, and n prime to A, and after suppressing these factors, let the quotients be A', B', then the greatest common measure M of the two latter quantities is that of the given quantities. Let M be contained in A' and B' respectively, p and q times, then and therefore A'=p M, B' = q M; A=mA=mp M, B=nB =ng M. But m and n, being factors of A and B, and respectively prime to B and A, are relatively prime, and so are P and q; also m and q are so, since q is a factor of B, and also n and p for a similar reason; hence m and p being relatively prime to n and q, mp and ng are relatively prime (154); therefore M is the greatest common measure of A and B. (106.) If a common factor of two given quantities be suppressed, the greatest common measure of the remaining quantities, multiplied by this factor, will be equal to that of the given quantities." Let c be a common factor of A and B, and let A = c A', Bc B'; then if M' be the greatest common measure of A', B', and M that of A, B, then is M = c M'. For let A'm M', and B' =n M', then is Amc M', and B = nc M' ; and since m, n, are relatively prime (104), therefore c M' is the greatest common measure of mc M' and nc M', or of A and B. That is, M = c M'. (107.) If either of two given quantities be multiplied by a factor which is prime to the other, the greatest common measure of the product and the other given quantity is the same as that of the given quantities." Let A, B, be the given quantities, and r a quantity prime to A, the greatest common measure of A and r B is the same as that of A and B. For let M be the greatest common measure of A and B, so that A = =mM, Bn M, and hence r Brn M. Now, since m and n are prime relatively, and also m and r, for r is prime to A, therefore m is prime to rn (152); and hence M is the greatest common measure of m M and rn M, or of A and r B. This theorem is a particular case of that in article (105), namely, when m in it is = 1. EXERCISES. 1. Find the greatest common measure of x3 — a2x and 23 — a3 2. Find the greatest common measure of 23- c2x and x2+2 cx + c2 3. Find the greatest common measure of a2 and a3 — a2x + 3 ax2 −3 x3 ·5ax+4x2 4. Find the greatest common measure of 6 a3-6 a2x +. 2 ax2 223 and 12 a2. 15 ax + 3x2 5. Find the greatest common measure of a2xa— a2y1 and x+x3y2 6. Find the greatest common measure of 6 a4. -5 a2x2 – 6×a and 4 a5 —— 6 a3x2 — 2 a2x2 +3x5 7. Find the greatest common measure of a8-28 and a19— x19 4x4 8. Find the greatest common measure of 625 11233-3x2 3x1 and 4 xa + 2 x3 - 18x2+3x 5... x2 + y2 3x2 7... 8... 2 x3 - 4x2+x-1 (108.) To find the greatest common measure of three quantities, find first that of two of them, and then that of this measure and the third quantity, and this last measure will be the one required." Let A, B, and C, be the three quantities, M the greatest common measure of A and B, and N that of M and C, then N is also that of A, B, and C. For any measure of A and B is one also of M, therefore any measure of A, B, and C, must be a measure of M and C; also any measure of M being also one of A and B, therefore any measure of M and C is also a measure of A, B, and C. Hence the greatest measure of M and C is also that of A, B, and C. (109.) The greatest common measure of four quantities is found in a similar manner, by first finding that of three of them, and then that of this measure and the fourth quantity; this last measure is the one required. And the greatest common measure of five, or of any number of quantities, is found in a similar manner." THE LEAST COMMON MULTIPLE OF QUANTITIES. (110.) A multiple of a quantity is any quantity that contains it exactly." And Thus, 6 is a multiple of 2 or of 3, and 24 of 2, 3, 4, &c. ; 12 as is a multiple of 12a, of 12aa, of aa, &c. 4 (ax) y2 is a multiple of 2 (a-x), of 2 y, &c. (111.) A quantity that contains two or more quantities is called a common multiple of them." Thus, 12 is a multiple of 3 and 4, or it is a common multiple of them. So 24 ax2 is a common multiple of 12, 8 ax, 6x2, &c. And 8 (x2-y2) 2 is a common multiple of 8 (2 y2), 4x2, &c. I. To find the least common multiple of two quantities. CASE I. When the quantities are simple. (112.) The least common multiple of the literal parts of any two simple quantities contains the highest powers of the letters contained in them, and may be found by inspec tion. The least common multiple of two quantities may also be found by dividing their product by their greatest common measure, or by dividing one of the quantities by their greatest common measure, and then multiplying the quotient by the other; and in the same manner, the least common multiple of two numbers may be found. If the quantities have numerical coefficients, their least common multiple must be prefixed to that of the literal parts.1 EXAMPLES. 1. Find the least common multiple of 4 a2xy3 and 6 æay3 Let M= the greatest common measure, and L = least multiple, 4a2xy3×6x4y5 2xy3 or=2a2x But the least common multiple may also be found by inspection. Thus, the highest given powers of all the letters are a2, xa, and y5, and the least common multiple of the literal part is a2x+ys. 2. Find the least common multiple of 622y3 and 8a2x526 6x8 L= a2x2у3z6—24 a2x2у3z6 3. Find the least common multiple of 12a5c8a6 and 18a3x5 12 x 18 - a5c8x6 = 2 × 18 a5c8x6 = 36 а3ç3x6 L= 6 EXERCISES. 1. Find the least common multiple of 12a4x6 and 18a2x2y*, and also of 10 a2cy2 and 15 ac2y6z2 2. Find the least common multiple of 18 and 24 a2x22, and also of 25 axy and 30 a2x23 10 3. Find the least common multiple of 6ac426 and 18c2-1o, and of 5 ac2x and 8 a3c34 4. Find the least common multiple of 120 axz5 and 64 *xy⭑z ANSWERS. 1...36axy and 30a2c4y62 3...18ac4z10 and 40a13xz1 2...72a2x4y52 and 150aa23 4...960axyz CASE II. When one or both of the given quantities are compound. (Î13.) 'Divide the product of the two quantities by their greatest common measure; or divide one of the quantities by the greatest common measure, and multiply the quotient by the other quantity.' EXAMPLES. 1. Find the least common multiple of 4aa2 and 3 (a — x) Here evidently M = 1, therefore L = 12 ax2 (a — x) 2. Find the least common multiple of 6 a2y (ax) and 4a (a2x2) M=2a (a-x), therefore L = 3 ay × 4a (a2x2) = 12 a2y (a2x2) 3. Find the least common multiple of 4 a (a2x2) and 6a2 (a3-x-3) M=2(a-x), therefore L=2 a (a + x) × 6x2 (a3 —x3) = 12 ax2 (a+x) (a3 — x3) — 12 ax2 (a1 + a3x · · аx3 — xo1) (114.) The rule for finding the least common multiple of two quantities is easily derived thus: Let A and B be two quantities, and M their greatest common measure, and let Am M, and B = n M, then (104) m, n, are relatively prime, or have no common factor, and their least common multiple is m n (157); therefore the least common multiple of m M and n M is mn M, but m M.nM A. B =L M 1. Find the least common multiple of 6a4x2y and 8 a2 (a + x) 2. Find the least common multiple of 8 a2 (a2-y2) and 12 ax4 3. Find the least common multiple of 4 a2 (a2 — x2) and 6 ax1 (aa—24) 4. Find the least common multiple of 6 x2 (a —x) and 4xy (a-x)2 (a + x) 5. Find the least common multiple of 8 ax (a3 + x3) and 12 a* (a + x)2 6. Find the least common multiple of 24 a6 (a3 — x3) and 5 a2x2 (a + x) (a — x) |