两个数的和是432,最大公约数加上最小公倍数和是7776求两个数的乘积
1个回答

两数最大公约数M,令两数为AM、BM,AB互质,最小公倍数ABM

AM+BM = (A+B)M = 432=2^4×3^3 ……①

ABM + M = (AB+1)M = 7776=2^5×3^5 ……②

则② / ①得

(AB+1) / (A+B) = 2*3^2 = 18

AB + 1 = 18A + 18B

等式右为偶数,则AB奇数,则A、B必然同为奇数.

(B-18)A = 18B - 1

A = (18B - 1)/ (B - 18) = (18B - 324 + 323)/(B-18) = 18 + 323/(B-18)

323 = 17 * 19 能被B-18整除,则有

①

B-18 = 17,B = 35,A = 37

或

B-18 = 19,B = 37,A = 35

M = 432 / (35+37) = 6

两个数就是 35*6、37*6

两个数的乘积 = 35*6 * 37*6 = (7776-6) * 6 = 46620

②

B - 18 = 323,B = 341,A = 1

M = 432/(323+1) 不为整数,舍弃

综上,两个数是35*6=210、37*6=222,其乘积为46620

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