Unit digit.....please help

You have three factors in this product and you are only interested in the units digit, so let's start by analyzing the units digit of each of the three factors:
a. 13^4 - since you are only interested in the units digit of this one, then you are looking for the units digit of 3^4. Since 3^2 = 9 and 3^4 = (3^2)^2, then the units digit of 3^4 and 13^4 will be the units digit of 9^2 = 1.
b. 17^2 - again, you are looking for the units digit of 7^2, which will be 9 (since 7^2 = 49)
c. 29^3 - units digit of 9^3 will be 9, since the units digit of 9^2 is 1 (9^2 = 81). Multiply 1 with 9 and you get 9.

So you have the following units digits: 1, 9 and 9. Multiply these babies and the number will have a final units digit of 1 (since, again, 9*9 = 81).

Yes, there is a pattern when it comes to multiplying numbers by itself.

2,3,7 and 8 - repeats its unit digit every four times. In other words, 2^1 will have the same units digit as 2^5. 3^2 will have the same as 3^6.

A quick way to determine the units digit for a number like 7^67 is to take 67 and divide it by 4. The remainder, 3, will be have the same units digit as 7^3. Another example, 2^38 will have the same units digit as 2^2.

0,1,5 and 6 will have the same units digit.

4 and 9 alternates between two numbers. 4 will be either 4 or 6. 9 will be either 9 or 1.