①
X是一位数时
2X = 20+X = 21 + (X-1)
X2 = 10X + 2 = 7X + (3X+2)
则3X + 2 = X-1 + 7K
2X = 7K - 3,显然K = 1、3,X = 2、9
②
X是二位数时
2X = 200+X = 196+ (X+4)
X2 = 10X + 2 = 7X + (3X+2)
则X+4 = 3X + 2 - 7K
2X = 7K + 2,显然K是偶数K=2T,X = 7T+1,即X是被7除余1的二位数：
X从15到99共13个.
③
X是三位数时
2X = 2000+X = 2002+ (X-2)
X2 = 10X + 2 = 7X + (3X+2)
则X - 2 = 3X + 2 - 7K
2X = 7K - 4,显然K是偶数K=2T,X = 7T-2,即X是被7除余5的三位数：
X从103到999共129个.
④
X是四位数时
2X = 20000+X = 19999+ (X+1)
X2 = 10X + 2 = 7X + (3X+2)
则X + 1 = 3X + 2 - 7K
2X = 7K - 1,显然K是奇数K=2T+1,X = 7T+3,即X是被7除余3的小于等于2009的四位数：
X从1004到2005共144个.
综上,共有2+13+129+144=288个.