By J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

ISBN-10: 3540665463

ISBN-13: 9783540665465

This quantity comprises the accelerated models of the lectures given by way of the authors on the C. I. M. E. educational convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers gathered listed below are wide surveys of the present examine within the mathematics of elliptic curves, and in addition include a number of new effects which can't be stumbled on somewhere else within the literature. due to readability and magnificence of exposition, and to the historical past fabric explicitly incorporated within the textual content or quoted within the references, the amount is definitely fitted to learn scholars in addition to to senior mathematicians.

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**Example text**

Remember that division by zero is undeﬁned! Dividing Integers 1. The quotient of two positive integers or two negative integers is the quotient of their absolute values. 2. The quotient of a positive integer and a negative integer (or a negative and a positive) is the opposite of the quotient of their absolute values. 3. The quotient of zero and any nonzero integer (zero divided by any nonzero integer) is zero. The next example illustrates this description of division. EXAMPLE 4 Find the quotient of the integers.

EXAMPLE 4 ■ Find the greatest common factor of 24 and 35. Solution 24 ϭ 2 # 2 # 2 # 3 ϭ 23 # 3 35 ϭ 5 # 7 Because there are no common prime factors, the greatest common factor is 1. ▼ PRACTICE YOUR SKILL Find the greatest common factor of 49 and 80. ■ The concept of “greatest common factor” can be extended to more than two numbers, as the next example demonstrates. EXAMPLE 5 Find the greatest common factor of 24, 28, and 120. Solution 24 ϭ 4 # 6 ϭ 2 # 2 # 2 # 3 ϭ 23 # 3 28 ϭ 4 # 7 ϭ 2 # 2 # 7 ϭ 22 # 7 120 ϭ 10 # 12 ϭ 2 # 5 # 2 # 6 ϭ 2 # 5 # 2 # 2 # 3 ϭ 23 # 3 # 5 The only factor that is common to all three numbers is 2.

3 3 3 3 3 3 3 a ba b a ba b 4 2 4 2 4 4 2 3 3 9 ϭ ± ≤ ± ≤ ϭ ϭ a ba b ϭ ϭ 2 2 3 1 4 2 8 3 2 a ba b 3 3 2 3 2 Notice that this is a form of 1 and 3 2 is the reciprocal of 2 3 3 3 2 3 divided by is equivalent to times . 5 Division of Rational Numbers If b, c, and d are nonzero integers, and a is any integer, then c a a Ϭ ϭ b d b # d c c a c a d by , we multiply times the reciprocal of , which is . The c b d b d following example demonstrates the important steps of a division problem. Note that to divide EXAMPLE 8 Find the quotient.

### Arithmetic theory of elliptic curves: Lectures by J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

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