by stevenH » Tue Nov 14, 2006 11:36 pm
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.