Number of $3$s in the units place

Question

I'm stuck on this problem:

(1) In the sequence $7,7^2,7^3,7^4,\ldots,7^{2014}$ how many terms have $3$ as the units digit?

After some random stuff, I have found that the unit digits of $7$ go in the order $7,9,3,1$ And then back to $7$. But I don't know how to utilize this in the problem!

Answer 1

You are doing it correctly. Simply divide $2014$ by $4$, where you get $503$, plus a $7^{2013}, 7^{2014}$ at the end. Since the $3$ comes $3rd$ in the sequence, it will not appear in the last two, so the answer is just $503$.