When a positive integer N is written in base 7, it is a three digit number.
When N is written in base 9, it is a three digit number that consists of the same digits but in reverse order.
Find all such decimal numbers N. Give the sum of all of these numbers in base 10.
The answer is 248.
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1 solution
Moderator note:
Good clear explanation of finding solutions to this diophantine equation.
The number written in base 7 is 49A+7B+C.
The number written in base 9 is 81C+9B+A.
These two numbers are equal so 49A+7B+C=81C+9B+A.
Simplifying and rearranging we get that 3A-5C=B/8 which implies that B is either 8 or 0.
Base 7 only uses the digits 0 to 6 so B has to be 0 and we are left with 3A-5C=0 and conclude that A=5 and C=3. The number is 503 base 7 or 305 base 9.
5x49+0x7+3x1=248.