Let m = 77ⁿ. The index n is given a positive integral value at random. What is the probability that the value of m will have 1 in the units place?

The units digit of 7ⁿ follows a cycle of 4: 7, 9, 3, 1 for n = 1, 2, 3, 4 respectively. Since 77ⁿ has the same units digit as 7ⁿ, the units digit is 1 when n is a multiple of 4. Out of every 4 consecutive values of n, only one gives units digit 1. Therefore the probability is 1/4.