Calculation of Unit Digit for (54)^89 - (49)^53

Step 1: Finding the unit digit of (54)^89
- The unit digit of 54 is 4. We need to find the unit digit of 4 raised to the power of 89.
- To find the pattern, we can calculate the unit digits for a few powers of 4:
4^1 = 4, 4^2 = 16, 4^3 = 64, 4^4 = 256, 4^5 = 1024, 4^6 = 4096, and so on.
- We can see that the unit digit of 4 repeats in a pattern of 4, 6, 4, 6, ... So, the unit digit of 4^89 will be the same as 4^1, which is 4.

Step 2: Finding the unit digit of (49)^53
- The unit digit of 49 is 9. We need to find the unit digit of 9 raised to the power of 53.
- To find the pattern, we can calculate the unit digits for a few powers of 9:
9^1 = 9, 9^2 = 81, 9^3 = 729, 9^4 = 6561, 9^5 = 59049, 9^6 = 531441, and so on.
- We can see that the unit digit of 9 repeats in a pattern of 9, 1, 9, 1, ... So, the unit digit of 9^53 will be the same as 9^1, which is 9.

Step 3: Subtracting the two unit digits
- The unit digit of (54)^89 is 4, and the unit digit of (49)^53 is 9.
- Subtracting 9 from 4, we get 4 - 9 = -5.
- Since we are interested in the unit digit, the negative sign is not considered. So, the unit digit obtained is 5.
Therefore, the unit digit obtained for (54)^89 - (49)^53 is 5.