Units Digit

Hi everybody!

I got a question concerning the pattern of getting the units digit in questions like the following:

What is the units digit of 3^17 × 17^54 × 135^19 × 31^21 × 42^17?

Is it true that the units digit is repeated with every 4th power, i.e. 3^2 has the same units digit as 3^6 and 3^10?

My approach:

3^17 ---> 17 divided by 4, remainder 1---> 3^1 ---> Units digit 3
17^54 ---> 54 divided by 4, remainder 2---> 7^2 ---> Units digit 9
135^19 ---> 19 divided by 4, remainder 3---> 5^3 ---> Units digit 5
31^21 ---> 21 divided by 4, remainder 1---> 1^1 ---> Units digit 1
42^17 ---> 17 divided by 4, remainder 1---> 2^1 ---> Units digit 2

3 x 9 x 5 x 1 x 2 = 270

Answer: Units digit = 0.

Would this be a good approach?