6 is 1 mod 5. So the sum of all the digits in base 6 of such a number, must be 0 or 1 mod 5 (because the last digit in base 5, has the value of the number mod

6 is 1 mod 5. So the sum of all the digits in base 6 of such a number, must be 0 or 1 mod 5 (because the last digit in base 5, has the value of the number mod 5, and here has to be 0 or 1) If a number only has 0's or 1's in a base, the sum of its digits in that base is just the number of 1's. Similarly, 5 is 1 mod 4, 4 is 1 mod 3, 3 is 1 mod 2. Inspect 82000: 10111000 in base 5: 4 1's, is 0 mod 4 110001100 in base 4: 4 1's, is 1 mod 3 11011111001 in base 3: 8 1's, is 0 mod 2 Seems worth mentioning.