The units digit is the last digit.

If we multiply any number ending in 3 by a number ending in 3,
we get a number ending in 9, since 3x3=9

If we then multiply that number ending in 9 by a number ending 
in 3, we get a number ending in 7, since 9x3=27

If we then multiply that number ending in 7 by a number ending 
in 3, we get a number ending in 1, since 7x3=21

If we then multiply that number ending in 1 by a number ending 
in 3, we get a number ending in 3, since 1x3=3.

Then the pattern starts all over again.

So if we continue t0 multipy 593x593x593x593x..., we continue

to get the pattern of last digits 3,9,7,1,3,9,7,1,....  So we 
just need to divide 143 by 4 to see how many times the cycle 
of last digits 3,9,7,1,3,9,7,1,... we go through in multiplying 
a number ending in 3 by itself.

   35
4)143
  12
   23
   20
    3

So the quotient 35 tells us that the last digits would cycle
around through the pattern 3,9,7,1, 35 times and the 3 remainder 
tells us that it would ends with a 7, since 7 is the third member 
of the cycle 3,9,7,1.

Answer: 7

Edwin