By Ivan Morton Niven, Herbert S. Zuckerman, Hugh L. Montgomery

The 5th variation of 1 of the normal works on quantity idea, written by way of internationally-recognized mathematicians. Chapters are rather self-contained for better flexibility. New positive factors comprise improved therapy of the binomial theorem, strategies of numerical calculation and a bit on public key cryptography. includes a superb set of difficulties.

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**Extra resources for An Introduction to the Theory of Numbers, 5th Edition **

**Example text**

For example, 5 is a prime in ~, for in the first place, 5 cannot be factored into real numbers in ~. 2). Thus, 5 is a prime in ~, and a similar argument establishes that 2 is a prime. We are now in a position to show that not all numbers of ~ factor uniquely into primes. Consider the number 10 and its two factorings: 10 = 2 . 5 = (2 + v'=6"")(2 - v'=6""). The first product 2 . 5 has factors that are prime in ~, as we have seen. Thus we can conclude that there is not unique factorization of the number 10 in ~.

6409 - 22 . (42823 - 6 . 6409) = ( -22) ·42823 + 147· 6409. In general, we set Sj+ I = 0, Sj = 1 and determine the numbers Sj_I' successively by the relation Si_1 = -qisi + Si+l. Put t i = Si+lri_1 + Siri. Since Sj-2'···' So it follows that the value of t i is independent of i. (b, c). The advantage of this method is we need construct only the one sequence {Si}' whereas in our former method we constructed two sequences, {uJ and {vJ The disadvan- Notes on Chapter 1 45 tage of this new method is that all the qi must be saved, as the Si are computed in reverse order.

In any positive integer, such as 8347, the last digit is called the units digit, the next the tens digit, the next the hundreds digit, and so forth. In the example 8347, the units digit is 7, the tens digit is 4, the hundreds digit is 3, and the thousands digit is 8. Prove that a number is divisible by 2 if and only if its units digit is divisible by 2; that a number is divisible by 4 if and only if the integer formed by its tens digit and its units digit is divisible by 4; that a number is divisible by 8 if and only if the integer formed by its last three digits is divisible by 8.