What is the units digit of 13^2003 ?

Look for a pattern

13^1 = 13
13^2 = (13)(13) = ---9 [aside: we need not determine the other digits. All we care about is the units digit]
13^3 = (13)(13^2) = (13)(---9) = ----7
13^4 = (13)(13^3) = (13)(---7) = ----1
13^5 = (13)(13^4) = (13)(---1) = ----3

NOTICE that we're back to where we started.
13^5 has units digit 3, and 13^1 has units digit 3
So, at this point, our pattern of units digits keep repeating 3, 9, 7, 1, 3, 9, 7, 2, . . .
We say that we have a "cycle" of 4, which means the digits repeat every 4 powers.

So, we get:
13^1 = --3
13^2 = ---9
13^3 = ----7
13^4 = ----1
13^5 = ----3
13^6 = ---9
13^7 = ----7
13^8 = ----1
13^9 = ----3
13^10 = ----9
etc.

Notice that when the exponent is a MULTIPLE of 4 (4, 8, 12, 16, ...), the units digit will be 1
Since 2000 is a MULTIPLE of 4, we know that the units digit of 13^2000 will be 1
Continuing with the pattern:
13^2001 = --3
13^2002 = ---9
13^2003 = ----7


Answer: C

Here's an article I wrote on this topic (with additional practice questions): https://www.gmatprepnow.com/articles/un ... big-powers

Cheers,
Brent