jmc

Question

For what base-6 digit d is 2dd5_6 divisible by the base 10 number 11? (Here 2dd5_6 represents a base-6 number whose first digit is 2, whose last digit is 5, and whose middle two digits are both equal to d).

Accepted Answer

We write 2dd5_6 in base 10 to get 2dd5_6=2 6^3+d 6^2 +d 6 + 5= 437 + 42d. We can subtract 39 11 from this quantity without changing whether it is divisible by 11. This subtraction yields 437 + 42d-429 = 8 + 42d. We can subtract 33d from this quantity, again not changing whether it is divisible by 11, leaving 8+9d. Now we try the possible values d=0,1,2,3,4,5 for a base-6 digit, and we find that only d=4 results in a number which is divisible by 11.