By Arjeh M. Cohen
The 1st interactive direction masking first and moment yr algebra. ranging from such basic subject matters as integers and divisions, modular mathematics and polynomials the content material extends to earrings, fields and permutation teams. The hypertext is written in Java-enhanced HTML, and Java applets illustrate the speculation whereas additionally contributing interactive calculators for computing with integers, polynomials and diversifications. the pc algebra approach hole is built-in all through, permitting the calculation and manipulation of mathematical items. additionally, collections for Mathematica notebooks and Maple worksheets evaluate the algorithms provided. a number of selection workouts offer clients with quick suggestions, whereas amenities for tracking scholars and a bulletin board whole this electronic path.
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The musical pitch of each note corresponds to its frequency, which is expressed in Hertz. If you double the frequency, you find a note an octave higher. If you change the frequency by a factor 3/2, you obtain a note which is a so-called fifth higher. , octaves higher. , fifths higher. Show that these two series of notes have no note in common, except the note you started with. 40. Suppose that a and b are coprime positive integers and that the positive integer n is a multiple of both a and b. Show that n is a multiple of a· b.
Determine the set of all integers that can be expressed in the form x· a + y· b + z· c with x, y, and z integers. 22. Determine the gcd of each of the following pairs of numbers, and write each gcd as a linear combination of the given numbers: 1. 5672, 234; 2. 5311, 121; 3. 32125, 1012. 23. Suppose a is a rational number such that 45· a and 36· a are integers. Is a necessarily an integer? And what if 20· a is also known to be an integer? 24. Find all integer solutions x and y to the following Diophantine equations.
Factorization of much larger numbers is exceptional. For example, there are numbers with more than 200 digits that have been factorized. One of the more famous examples is the number called RSA-129. K. Lenstra, et al. was announced. RSA-129: 1143816257578888676692357799761466120102182967212423625625618429 35706935245733897830597123563958705058989075147599290026879543541 = 3490529510847650949147849619903898133417764638493387843990820577 × 32769132993266709549961988190834461413177642967992942539798288533 It is not difficult to check that the product of these two factors is indeed the large number: any computer system that can work with these large numbers will confirm it.
Algebra Interactive by Arjeh M. Cohen