Raising to large powers

[NickR]

There was a problem on a recent test: "What is the tens digit of 17^2006?" with the five answer choices: 1,3,5,7,9. I used the euler's totient function way and got an answer of 6. This wasn't a choice on the test, so I went home and crunched out the actual number on Mathematica and got 6 also...so what's up with this problem? Is there something I'm doing wrong or not seeing?

Thanks in advance,
Nick

[CQuary]

This is the second time that I have seen a question asking for the tens digit. Both times the answer has been the ones digit. I don't know if the questions are meaning to ask about the ones digit or not but I have seen a couple of errors on the tests this year. My UIL coordinator found three mistakes(including this one) on the math test answer key today(costing me 16 points) and last week there were two or three others. Oh well...that happens.

[NickR]

Yeah, I've noticed a few more mistakes also...but of course it would be impossible for the writers to catch every mistake. I was just wondering if I had missed something obvious on this problem.

[AuSmith]

Yea, I had to lower my expectations on UIL/TMSCA reliability (something like this should have been caught fairly easily). I mean, the tests I wrote probably have such errors, and if they didn't, it doesn't pay well enough to check thoroughly. I think I heard once that no more than one other person checks the tests for accuracy. I would have really appreciated more people looking tests over before the final product. Though, there are a number of other accurate competitions. Olympiad contests run through much scrutiny before the final. In fact I can't recall having a problem with an olympiad or a Putnam.

[stevenH]

I apologize for the errors on the last few tests; apparently I don't know the difference between the tens digit and ones digit ;). Do you know what other questions had errors? I'll make sure to correct them for future TMSCA test pool orders. A little secret: only the test writer looks over the tests, so it's very easy for me to overlook typos, even after triple-checking.

You are absolutely right, the answer should have been 6. My intention was to say "units digit", and I had in mind a slightly easier problem. However, for future tests, I will definitely consider having tens digit and hundreds digit problems and rest assured the correct answer choice will be there.

As you said, you can use Euler's totient theorem, i.e., a^(phi(n)) = 1 (mod n) for relatively prime a and n, where phi(n) is the number of positive integers relatively prime to n and less than n. We are interested in the tens digit, so we choose n = 100. Since phi(100) = (2^2-2^1)*(5^2-5^1)=40, we have a^40 = 1 (mod 100) for relatively prime a and n. Therefore, 17^2000 = (17^40)^50 = 1 (mod 100) and 17^2006 = 17^6 = 69 (mod 100). Thus a tens digit of 6.

[NickR]

Thanks for your reply : ). Anyway, I only noticed a couple more or so on the #3 test of this year. On problem #3, the answer is E(7) however I'm pretty sure that the answer is D(6). They even counted it correct at the meet, so perhaps they caught it. Last one I saw was problem #31...shouldn't the answer be B(-tan x) since sin(x+pi) = -sinx and cos(x+2pi)=cosx, then (-sinx)/(cosx)=-tanx?