perplexus.info :: Just Math : Units Digit Determination

Units Digit Determination (Posted on 2015-01-07)

N is a 9-digit duodecimal (base 12) palindrome such that:

The first two digits of N are not consecutive, and:
The sixth and seventh digits of N are not consecutive

What is the units digit of the sum of all possible values of N?

How about N being a 9-digit hexadecimal (base 16) palindrome instead?

*** N is of course positive and does not contain any leading zero.

See The Solution Submitted by K Sengupta

Rating: 4.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

Simple analysis (spoiler?)

| Comment 1 of 3

Well, the final digit can be any value from 1 to 11 (i.e. the base - 1).

The average value is 6 (ie, the base/2).

So the units digit of the sum of all possible N is is 0 if the number of possibilities is even and 6 if it is odd.

The number of possibilities = (# of possible first two digits)*(number of possible 3rd and 4th digits)*(number of possible middle digits).

The number of possible middle digits is even, so the total number of possibilities is even, so the units digit of the sum is 0.

The same argument and answer applies to any even base, including 16.

I wonder if I have made a mistake. If so, somebody will tell me. Probably Charlie's computer, which I can hear adding up a lot of 9 digit numbers right now.

Posted by Steve Herman on 2015-01-07 15:01:34

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