We start from a problem: Problem: Find the number of elements in such that it is congruent to module for some integer . Solution. The problem is the same as finding the number of solutions to     Using Chinese Remainder Theorem, the problem reduces to the system of equations         Our

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There is a problem in Real Analysis by Halsey L. Royden, Patrick M. Fitzpatrick. Suppose that outer-measure is defined by covering sets by countable collections of closed, bounded intervals rather than coverings by open, bounded intervals. Show that the outer- measure remains unchanged. I give my answer here. Proof. Let be a set of real

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