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Globalizing Interests: Pressure Groups and Denationalization
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Lea", 6' = kb", and so a = hka",b = hkb" and a, b have the common factor hk which is greater than h. e. xa — yb = h, showing that the greatest common factor is expressible as the difference of a multiple of a and a multiple of b. I t follows that every number which divides both a and b also divides h, so that h is the greatest common factor in this sense also. We shall illustrate a method of finding values of x and y such that xa — yb = 1 by means of examples. ) Starting with 5 in the upper row, we pass to 10 in the line below, then look for 10 in the upper row and pass to 4 below it, continuing 53 N U M B E R S FOB COUNTING in this way through the sequence 9, 3, 8, 2, 7, 1, and stopping when we reach 1.
Recently a number as gigantic as Si945 was shown to have the factor 5 x 21947 + 1. F. Gauss, then a boy of 18, and destined to become the greatest mathematician of all time, discovered a connection between the Fermat numbers and the problem of dividing the circumference of the circle into equal parts using only ruler and compasses. Gauss discovered that the circumference can be divided into n parts if and only if n is a power of 2, or a Fermat prime or a product of a power of 2 and distinct Fermat primes (calling 3 the Fermat prime $0 f° r a reason to be discussed later).
Until we reach 68 + 7 = 75, showing that 7 is the remainder when 75 is divided by 17, and since 68 = 4 x 17, therefore 4 is the quotient. Division bears the same relation to subtraction that multiplication bears to addition; that is to say division is equivalent to repeated subtraction. The question "what is the quotient when 75 is divided by 17" is the same as the question "how many times can 17 be subtracted from 75". For (75 - 17) - 17 = 75 - (17 + 17) - 75 - 2 x 17 (75 - 2 x 17) - 17 = 75 - (2 x 17 + 17) - 75 - 3 x 17 (75 - 3 x 17) - 17 = 75 - (3 x 17 + 17) = 75 - 4 x 17 = 7.
Encyclopaedie der mathematischen Wissenschaften und Anwendungen. Algebra und Zahlentheorie
by George
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