SOLUTION: Can you please help to answer the following question: . 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-d

Question 12200: Can you please help to answer the following question:
.
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.
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
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

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