A Minor doubt

Hi, there. I'm happy to help with this.

First of all, the patterns of what the unit digits of powers are only hold for positive integer powers, powers n > 0. Any integer to the power of 0 is 1, and at n = 0, that pattern trumps whatever individual pattern the individual base has.

As for powers of 4, you are perfectly correct: the unit digits of the powers are 4, 6, 4, 6, 4, 6 ---> 4^(odd power) has a unit's digit of 4, and 4^(even power) has a unit's digit of 6. Thus, 100 is even, so 4^100 would have a units digit of 6.

I am holding the Manhattan GMAT book to which you referred, and I believe the question you cited is p. 66, #12: "If Sn = 4^n + 5^(n+1) + 3, what is the units digit of S100?" That question ---- not simply the question about the units digit of 4^100 by itself ---- has an answer of 4. The reason is: the 4^100 part would have units digit of 6, any power of 5 has a units digit of 5, and 6 + 5 + 3 = 14, something with a units digit of 4.

Does that all make sense? I hope that clears up your confusion. Please let me know if you have any further questions.

Mike