Euclid’s Algorithm for the Greatest Common Divisor

People have been using numbers, and operations on them like division, for a very long time for practical purposes like dividing up the money left by parents for children, or distributing ears of corn equally to groups of people, and more generally to conduct all sorts of business dealings. It may be a bit of a surprise that things like calculating divisors of numbers also form the core of today’s methods ensuring security of computer systems and internet communications. The RSA cryptosystem that is used extensively for secure communications is based on the assumed difficulty of calculating divisors of large numbers, so calculating divisors is important even today. A related and even more basic notion is that of multiples of quantities. A natural way to compare quantities is to “measure” how many times we need to aggregate the smaller quantity to obtain the larger quantity. For example, we may be able to compare two unknown lengths by observing that the larger length can be obtained by “aggregating” the smaller length three times. This provides a sense of how the two lengths compare without actually knowing the two lengths. The larger quantity may not always be obtainable from the smaller quantity by aggregating it an integral number of times. In this scenario, one way to think would be to imagine each of the two quantities to be made up of smaller (identical) parts such that both the quantities can be obtained by aggregating these smaller parts an integral number of times. Obviously, we will need a greater number of these parts for the larger quantity than for the smaller one. For example, when comparing two weights, one might observe that the larger one can be obtained by aggregating some weight 7 times whereas the smaller weight can be obtained by aggregating the same weight 5 times. This provides a basis for comparing the two weights. Of course, in the above scenario, one can also observe that if we chose even smaller parts to “split” the weights (say a quarter of the first one), the first weight would be obtained by aggregating this even smaller weight 28 times and the smaller of the two original weights would be obtained by aggregating this smaller part 20 times, which also provides us a sense of the relative magnitudes of