兩數(shù)最大公約數(shù)M,令兩數(shù)為AM、BM,AB互質(zhì),最小公倍數(shù)ABM
AM+BM = (A+B)M = 432=2^4×3^3 ……①
ABM + M = (AB+1)M = 7776=2^5×3^5 ……②
則② / ①得
(AB+1) / (A+B) = 2*3^2 = 18
AB + 1 = 18A + 18B
等式右為偶數(shù),則AB奇數(shù),則A、B必然同為奇數(shù).
(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,則有
①
B-18 = 17,B = 35,A = 37
或
B-18 = 19,B = 37,A = 35
M = 432 / (35+37) = 6
兩個數(shù)就是 35*6、37*6
兩個數(shù)的乘積 = 35*6 * 37*6 = (7776-6) * 6 = 46620
②
B - 18 = 323,B = 341,A = 1
M = 432/(323+1) 不為整數(shù),舍棄
綜上,兩個數(shù)是35*6=210、37*6=222,其乘積為46620