Question 12200
    When a positive integer N is written in base 9, it is a two-digit number. When 6N is written in base 7, it is a three-digit number obtained from the two-digit number by writting digit 4 to its right. Find the decimal representationsof all such numbers N. [/quote]

 Sol: By the given condtions, we have
 There are integers 1 <= a < 7, 0 <= b < 7 such that
 N = 9 a + b ... 1) 
6N = 7^2 a + 7 b + 4 ...(2)

By (1)*6: we have 6 N = 54 a + 6 b = 7^2 a + 5 a + 6 b = 7^2 a  + 7 b + 4.
 We have 5 a = b + 4 ...(3) 

Note, 1 <= a <= 6, 0 <= b <= 6 and 5 a = b + 4.

 Value of a starting from 1 ,consider the table below:

 a    b(=5a-4)   N = 9 a + b   6N   (6N)base 7
--------------------------------------------
 1	1        10          60     114 (OK)
 2	6        24         144     264 (OK)
 3    15-4 >6 (invalid)
 Similarly ,we see that b > 6 whenever 6>= a > 3,so no valid b if a > 3.


 Thus, we obtain the two possible solutions 10 or 24.

 [Double check : N = 10 [base 9] = 11, 6N = 60 , since 60 = 7^2 + 7 + 4,
 so 60[base 7] = 114. 
 N = 24 [base 9] = 26, 6N = 144 , since 144 = 2*7^2 + 6*7 + 4,
 so 144[base 7] = 264. 

 This question may be not quite easy for you. Try to read carefully
 about every step.

 Kenny