Suggest a shortest method to solve it

Unit's digit of 33^43 = unit's digit of (4*8 + 1)^43 = 3
Unit's digit of 43^33 = unit's digit of (4*10 + 3)^43 = 7

If 43 = 4*10 + 3 then why not 33 = 3*10 + 3? Then the answer would be 7 + 7 = 14 or 4 (C)

vikrambansal wrote:

Unit's digit of 33^43 = unit's digit of (4*8 + 1)^43 = 3
Unit's digit of 43^33 = unit's digit of (4*10 + 3)^43 = 7

If 43 = 4*10 + 3 then why not 33 = 3*10 + 3? Then the answer would be 7 + 7 = 14 or 4 (C)

3¹ --> units digit of 3.
3² --> units digit of 9. (Since the product of the preceding units digit and 3 = 3*3 = 9.)
3³ --> units digit of 7. (Since the product of the preceding units digit and 3 = 9*3 = 27.)
3� --> units digit of 1. (Since the product of the preceding units digit and 3 = 7*3 = 21.)
From here, the units digits will repeat in the same pattern: 3, 9, 7, 1.
The units digit repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 3 is raised to a power that is a multiple of 4, the units digit will be 1.

Thus:
33�� and 43³² each have a units digit of 1.
From here, the cycle of units digits will repeat: 3, 9, 7, 1...
Thus:
33�¹ and 43³³ each have a units digit of 3.
33�² has a units digit of 9.
33�³ has a units digit of 7.

Result:
Since n = 33�³ + 43³³, n ---> (units digit of 3) + (units digit of 7) = units digit of 0.

The correct answer is A.

Thanks to both of you. I realized that it's not derivation of no. 33 (10*3 + 3 or 8*4 + 1) but POWER 33.

Thanks all. I fully understand after the breakdown. This is a great example of a deeper question that comes to mind on quant questions like this. What would give you the clues or the signals to think to break it down? What should one be thinking, or what's the thought process to know how to unpack a question like this?