Remainder Problem

Find the sum of the remainders obtained when a number n is divided by 9 and 7 successively, if n is the smallest number that leaves respective remainders of 4, 6 and 9 when divided successively by 13, 11 and 15.

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Number of ordered Pairs

Let S be a set of n consecutive natural numbers.
How to find the number of ordered pairs (A,B), where A and B are subsets of S and A is a proper subset of B?

For Example:
Let the set S = {1, 2, 3, 4, 5, 6, 7, 8}.
Find the number of ordered pairs (A, B), where A and B are subsets of S and A is a proper subset of B.

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Number of paths

For a rectangular grid having a x b cells as shown in the below figure:-

Find the number of shortest paths along the edge of the cells. Starting from the bottom-left point and end at right-up point.
For simplicity they are marked as (0,0) and (a,b)
Write a pseudo code to print all such
paths.

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