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 goal, thus, is to find the number of solutions for each of the equation. We…

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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 numbers. , defined similarly with intervals to be closed. Now consider , we form the…

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