Hi Christopher.

Examine the sequence (n, units digit of 23ⁿ):
(0, 1), (1, 3), (2, 9), (3, 7), (4, 1)...now we are back to 1.

So it repeats: 1, 3, 9, 7, 1, 3, 9, 7, 1, ... The period is 4 units.
This means that the units digit of 23⁴⁵ is the same as the units
digit of 23^{45 - 44} = 23¹.
Thus the units digit of 23⁴⁵ is a three.

Use the same principle to discover the sequence and period for 67⁸⁹.
Remember that the 6 in 67 has no effect on the units digit.

Then add the two units digits together to find your final answer.

Hope this helps,
Stephen La Rocque.