10 Case

Math challenge .Try to solve it?

9.Find the number of integers N satisfying the following conditions:
A)1 < n < 2013
B)the last digit of n^99 equal to last digit of n

There are 10 cases to consider since all we care about is the last digit.

xxxx0 → all powers end in 0 so (10, 20, 30, .... 2010) → 201 numbers
xxxx1 → same deal ... (01, 11, ... 2011) 202 more
xxxx5 → ditto ... (05, .... 2005) ... 201 numbers
xxxx6 → once again (6 ... 2006) ... 201 numbers

Then it slows down:
Powers of 2 go ...
2, 4, 8, 6, 2 ... every 4th one ends in 2.
1, 5th, 9th, 13, ... 97th power ... that one doesn't work at all

Powers of 3 go
3, 9, 7, 1, 3 .... same deal there as with 2.

Powers of 4 go
4, 6, 4 .... every odd power ends in 4 so we get all those.
4, 14, ... 2004 ... → 201 numbers

Powers of 7 go
7, 9, 3, 1, 7 ... but again that's a loser since 99 != 1 (mod 4)

Powers of 8 go
8, 4, 2, 6, 8 ... no good on that one

Powers of 9 are
9, 1, 9, .... so those are all good ... (9 .... 2009)

Summarizing:
xxxx0 ... 201 numbers
xxxx1 ... 202 numbers
xxxx5 ... 201 numbers
xxxx6 ... 201 numbers
xxxx4 ... 201 numbers
xxxx9 ....201 numbers

2, 3, 7, 8 ... none

Total = 1207

(I hope got all the 201's and 202's straight ...
it's annoying the range is 1 to 2013 instead of a nice regular
0 to 2009 or 2019 ... it adds an unnecessary complication to it.)

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