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## Why Is Fraction 1360518655681506310/99999999999999999999 Special?

For this problem, stop and think before rushing into possible solution attempts. Some calculations may be necessary, but think first...and after these calculations.

[Q] If N is a random whole number, what is the probability that the last digit of the sum of the first N whole numbers is a 7?

Again, stop and think...a lot.

Source: Mathematics Magazine, 1957, pp. 290-291.

Hint: Systematically, explore sums for different N-values.

Solution Commentary: For N = 0, 1, 2, 3, 4, 5, ...., you should obtain sums with these last digits: 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0,.... The digits now repeat, which means that the probability of a 7 is zero. And, the above sequence is the repeating cycle for the fraction 1360518655681506310/99999999999999999999.