The concept given in manhattan is that any number divided by 10 has the remainder as its unit digit. so the question can be rephrased as finding the unit digit, but how do we do that?
Hint: Since 13 and 17 ends with 3 and 7 respectively, we have to find the pattern of units digits when 3 and 7 are raised by different powers.
Solution:
Consider 13^{17}:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
So the pattern is [3, 9, 7, 1]. The sequence repeats after 4 numbers. Since we have 13^{17}, find the closest number to 17 divisible by 4. That would be 16. 13^{16} would end with 1, so 13^{17} would end with 3.
Similarly, consider 17^{13}:
The pattern for 7 is [7, 9, 3, 1]. 17^{12} would end with 1 (12 is divisible by 4). So 17^{13} will end with 7.
Finally, 13^{17} + 17^{13} will end with 3 + 7 = 10
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