MATHEMATICS THE EUCLIDEAN ALGORITHM EUCLIDS ALGORITHM EFFICIENT METHOD COMPUTING GREATEST COMMON DIVISOR LARGEST NUMBER DIVIDES LEAVING REMAINDER THE ANCIENT GREEK MATHEMATICIAN EUCLID EUCLIDS ELEMENTS EXAMPLE ALGORITHM PROCEDURE PERFORMING CALCULATION ALGORITHMS IN COMMON BE USED TO REDUCE FRACTION SIMPLEST CRYPTOGRAPHIC THE PRINCIPLE GREATEST COMMON DIVISOR OF TWO NUMBERS CHANGE DIFFERENCE SMALLER PROCESS PAIRS THE ORIGINAL REVERSING THE STEPS SUM OF ORIGINAL POSITIVE NEGATIVE INTEGER FACT ALWAYS BE IDENTITY VERSION ABOVE SUBTRACTION STEPS FIND THE GIVEN THE ALGORITHM SHORTCUTS DIVIDED BY STOPS REACHING ZERO NEVER TIMES THIS WAS PROVEN GABRIEL MARKS THE BEGINNING OF COMPUTATIONAL COMPLEXITY THEORY IMPROVING EFFICIENCY DEVELOPED THEORETICAL PRACTICAL DIVISION MODULAR ARITHMETIC COMPUTATIONS CRYPTOGRAPHIC PROTOCOL USED TO INTERNET COMMUNICATIONS BREAKING CRYPTOSYSTEMS FACTORING COMPOSITE NUMBERS SOLVE EQUATION FINDING SATISFY MULTIPLE THE CHINESE THEOREM CONSTRUCT CONTINUED FRACTION ACCURATE RATIONAL APPROXIMATIONS REAL NUMBERS TOOL PROVING THEOREMS NUMBER THEORY PRIME NATURAL NUMBERS GEOMETRIC LENGTHS GENERALIZED THE CENTURY GAUSSIAN INTEGER POLYNOMIAL MODERN ABSTRACT ALGEBRA NOTIONS EUCLIDEAN DOMAIN CALCULATES NATURAL NUMBER SYNONYMS GREATEST COMMON FACTOR HIGHEST COMMON FACTOR GREATEST COMMON MEASURE WRITTEN NOTATION MATHEMATICAL VECTORS SAID TO COPRIME RELATIVELY PRIME PROPERTY THEMSELVES PRIME NUMBER NEITHER PRIME FACTORS THIS IS TRUE THE NATURAL MUST COMMON FACTOR FACTORED C DIVIDE UNIQUE COMMON DIVISOR DIVISIBLE CONSIDER AREA RECTANGLE DIVIDED LENGTH GRID SQUARES VALUE EDGE THE OTHER THE PRODUCT PRIME FACTOR FACTORS HERE INSTANCE EMPTY PRODUCT IN OTHER WORDS ADVANTAGE COMPUTE FACTORIZATION INTEGERS BELIEVED COMPUTATIONALLY SECURITY CRYPTOGRAPHY SYSTEMS DEFINITION OF HELPFUL ADVANCED PARTICULARLY RING THEORY NONZERO INTEGRAL LINEAR COMBINATION POSITIVE NUMBER OF THE FORM THE SET LINEAR COMBINATIONS ACTUALLY IDEAL ALONE ELEMENT PRINCIPAL IDEAL IDEALS PRINCIPAL PROPERTIES IN FACT EASIER TERMS EQUIVALENCE DEFINITION DEFINITIONS CALCULATED TAKING GCDS ARBITRARILY SUCH THAT OUTPUT STEP INPUT THE NEXT ONE COUNTS INITIAL THE NEXT SO ON BEGINS REMAINDERS DECREASE EVERY STEP LESS THAN PREDECESSOR THE GOAL QUOTIENT THE EQUATION THE REMAINDER SMALLER THAN EQUAL THE NUMBERS SEQUENCE EQUATIONS THE FIRST STEP SWAPS FOR ALL NEVER BE EVENTUALLY POINT CANNOT INFINITE NONNEGATIVE INTEGERS VALIDITY IT MUST BE LESS THAN OR EQUAL TO SECOND STEP INCONSISTENT UNLESS ITERATING NONE LEAVE ANALOGOUS ARGUMENT SHOWS THE FIRST PART OF THE ARGUMENT REVERSE IT FOLLOWS LESS THAN THREE SEVEN THE LAST ENDS PRIME FACTORIZATION TABULAR TILING ANALOGY WE WISH COVER SQUARE TILES ATTEMPT TILE LEAVES RESIDUAL THE SEQUENCE THE SQUARE PREVIOUS THE DIMENSIONS ADJACENT FIGURE MAGNITUDE THE DEFINITION OF EUCLIDEAN DIVISION ALWAYS EXIST ORIGINAL VERSION THE PROCESS THUS MOREOVER QUOTIENTS REPLACE MODULO OPERATION SIMPLY PSEUDOCODE FUNCTION RETURN AT THE BEGINNING THE VARIABLE HOLDS THE LATEST EQUIVALENT RECURSION FORMULA TEMPORARY VARIABLE AT THE END THE LOOP CONTRARY ARBITRARY POSITIVE INTEGERS ELSE ALTERNATE HOLDING REDUCED RECURSIVE THE EQUALITY CONDITION THE EQUIVALENT TURN PREVIOUSLY ALTERNATIVE GETS VARIANT EUCLIDEAN ALGORITHM LEOPOLD KRONECKER FOR EVERY MINIMAL IF AND ONLY IF CHOSEN ORDER THE GOLDEN RATIO APPEARS BOOK BOOK IT LINE SEGMENTS SAY IT THERE FOR REAL NUMBER MEASURED UNITS NATURAL UNIT VOLUME THE CONCEPT UNKNOWN THAT TIME GREATEST PROBABLY DISCOVERED COMPILED RESULTS MATHEMATICIANS MATHEMATICIAN HISTORIAN VAN DER WAERDEN SUGGESTS TEXTBOOK WRITTEN BY THE SCHOOL PYTHAGORAS EUDOXUS OF CNIDUS EUDOXUS JUDGING TECHNICAL TERM RECIPROCAL CENTURIES INDIA ARISE ASTRONOMY INDIAN MATHEMATICIAN ASTRONOMER ARYABHATA PULVERIZER EFFECTIVENESS DIOPHANTINE EQUATIONS SPECIAL CASE ALREADY CHINESE MATHEMATICIAN SUN TZU THE GENERAL SOLUTION PUBLISHED QIN JIUSHAO MATHEMATICAL TREATISE IN NINE EUROPE SECOND EDITION LIKEWISE EXTENDED EUCLIDEAN ALGORITHM THE ENGLISH NICHOLAS SAUNDERSON ATTRIBUTED ROGER COTES CONTINUED FRACTIONS DEVELOPMENT OF NUMBER SYSTEMS EISENSTEIN INTEGER CARL GAUSS WORK GAUSS DISQUISITIONES ARITHMETICAE PETER GUSTAV LEJEUNE DIRICHLET SEEMS DESCRIBE BASIS LEJEUNE DIRICHLET HOLD TRUE FOR ANY SYSTEM LECTURES ON NUMBER THEORY EDITED RICHARD DEDEKIND STUDY ALGEBRAIC INTEGER GENERAL TYPE PROVE GAUSSIAN INTEGERS DEFINED NUMBER SYSTEM BELOW CLOSING DECADES ECLIPSED GENERAL THEORY OF APPLICATIONS CHARLES STURM USEFUL STURM CHAIN METHOD COUNTING THE REAL ROOTS INTEGER RELATION ALGORITHM RELATIONS BETWEEN NOVEL HELAMAN FERGUSON FORCADE COLE DAVIE GAME THE GAME OPTIMAL THE PLAYERS BEGIN PILE CONSIST PLAYER STONES THE WINNER LINEAR POSSIBLE THE ORDER BEGINNING THOSE TWO FORMULAE YIELDS REACHED AFTER ALL THEREFORE THE CONTEXT YET ANOTHER INTEGER MULTIPLE MEMBER CHOOSING CONVERSELY SEEN MULTIPLYING SUMS GENERATOR THE MODERN CONCEPTS PRINCIPAL IDEAL DOMAIN DOMAIN CERTAIN PROBLEMS MEASURING CUPS SECOND CUP VOLUMES ADDS VALUES APPROACH BY INDUCTION UP