How many positive integers not divisible by 6 have a base 6 representation which

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2 digit answers: 6x + y = 9y + x => 5x = 8y => no solution, bcause you can't have 8 in base 6 3 digit answers: 36x + 6y + z = 81z + 9y + x => 35x = 3y + 80z Both sides must be multiples of 5, so y must be 5: => 35x = 15 + 80z Right side must be a multiple of 7. 175 works. i.e. x=5, y=5, z=2 i.e. 552_base6 = 255_base9 Check this and you will see both numbers are 212 base 10 No other solutions because the next z which gives a multiple of 7, is greater than 6 You should then check 4 and 5 digit numbers in the same way: 216w + 36x + 6y + z = 729z + 81y + 9x + w Notice that z and w must be the same values in mod 3 aritmetic. And since w > z, you only need to check (w=4, z=1), and (w=5, z=2) There is no need to check 6 digit numbers, because the largest 6 digit base 6 number (6^6 - 1) is smaller than the smallest 6 digit base 9 number (9^5) so answer is 6

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