求自然数a1a2.an,使得12×2a1a2.an1=21×1a1a2.an2

Question

求自然数a1a2.an,使得12×2 a1a2.an 1=21×1 a1a2.an 2

Answer 1

两边可以先把3约掉,有 4 * 2 a1a2.an 1 = 7*1 a1a2.an 2 ①
2 a1a2.an 1 = 2*10^(n+1) + 10a1a2.an +1
1 a1a2.an 2 = 10^(n+1) + 10a1a2.an +2
则①转化为 8*10^(n+1) + 40a1a2.an +4 = 7*10^(n+1) + 70a1a2.an +14
∴30a1a2.an=10^(n+1)-10 = 10(10^n -1)
注意10^n -1就是99...9,一共n个9
∴a1a2.an=33.3 ,一共n个3