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T, Exercise: Partition a proper fraction into a sum of Egyptian fractions. Analysis: We propose to use the Fibonacci maximal algorithm()! to solve this problem . Su c h fra ctions were used by the ancie nt Egypti ans, beca use th ey lacked practi ca l method s for handling other types of fractions . ()IFibonacci: Leonardo da Pisa, known by the name of Fibonacc i, was bo rn in Pisa aroun d 11 75 and publi shed this algorithm in 1202 . EXERCISES USING INTEGERS Let us assume that you are given the fraction ~ to decompose.

10 illustrates this point. /f is used as a symbol for comparison. 10: Decision Points: 2 and 3 Exits - - - - - - - - - - - - - ' There are instances where a decision point in a flowchart could have more than three exits. This might happen because the flowchart must represent a general class of algorithms. 11 shows how numerous exits might be represented . 6 A "Flip-Flop" Technique for Branching How can we flowchart a loop so that the left side of the flowchart is executed on each odd passage through the loop and the right side is executed on each even passage?

11 shows that Armstrong numbers are not numerous. EXERCISES USING INTEGERS R is one the digits of I. We add R cubed to 5 (5 is the sum of the cubes of the previous digits). Q = KilO R=K-IO·Q S = S + R' K=Q NO When K - 0 we have gone through all the digits of 1. 3 Partitioning a Fraction into Egyptian Fractions A fraction that has a numerator of 1 is sa id to be an Egyptian fraction (2! (for example, 110 etc) . A fraction that has a numerator that is smaller than its denominator is called a proper fraction .