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Given a rectangle having sides in the ratio 1:phi, the golden ratio phi is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio 1:phi. Such a rectangle is called a golden rectangle.

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For more reading: http://mathworld.wolfram.com/GoldenRectangle.html

find the remainder when 3^2007 is divided by 7

ANS:6

solution: Let N = 3^2007= (3^3)^669 = 27^666 = (28-1)^669

expanding using binomial theorem   N= terms contaaining 28 -1(since 669 is odd)

thus,N =28k-1(k is integer), thus N+7=28k+6

when 28k+6 is divided by 7 remainder=6

thus N+7 gives remainder 6 when divided by 7

hence N gives remainder 6!

I was reading through and I found this, really good to think on, simple solution though: 

Lagrange’s Four-Square Theorem states that every positive integer can be written as the sum of at most four squares.  For example, 6 = 2² + 1² + 1² is the sum of three squares.  Given this theorem, prove that any positive multiple of 8 can be written as the sum of eight odd squares.

Reference: http://www.qbyte.org/puzzles/puzzle16.html#p159

This is to inform you that I am currently working with MAQ Software as Software

Development Engineer.

There are ‘n’ straight lines in a plane in a general position. No two parallel and no three concurrent. All of them intersect each other. Find the number of disjoint regions into which the plane is divided by these lines??               For eg: for 3 straight lines the number of disjoint regions is 7.    

Calculate for ‘n’= 64.

There are ‘n’ straight lines in a plane. No two of them are parallel and no three are concurrent. All of them intersect each other. There intersection points are joined in pairs. Find the number of new straight lines that are formed??

Calculate for ‘n’= 64.

Here is one of the best staircase problems. An elegant solution is expected to this one..

There is a staircase and a person intends to climb it. He can take either one or two steps at a time. In this way, in how many ways can he climb a 12 step staircase??

A magic square is an n x n matrix of the integers 1 to n² such that the sum of every row, column and diagonal is the same.

Problem Statement: Generate magic square for odd values of n

Download C code

Explanation:
Coxeter has given the following simple rule for generating a magic square when n is odd:
Start with 1 in the middle of the top row; then go up and left, assigning numbers in increasing order to empty squares; if you fall off the square imagine the same square as tiling the plane and continue; if a sqaure is ocupied, move down instead and continue.