1.10. INDEPENDENT PROJECTS 31 We now present the remaining case as an exercise. Exercise 1.10.38. Let p/q be a rational number in (0, 1) with q = 2a5br where r is relatively prime to 10. Let k = max(a, b) and let n be the smallest positive integer such that r divides 10n − 1. Show that, after k digits, the decimal expansion of p/q is periodic of length n. Exercise 1.10.39. Can any of the above decimal expansions terminate in all 9’s?

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