有a、b两个正整数,他们的和是432,它们的最大公因数和最小公倍数的和是7776,求a和b的乘积.

两数最大公约数M,令两数为a = AM、b=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 a=35*6、b=37*6 或交换顺序
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,ab两个数是210、222,其乘积为46620