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Prove that xn yn has x y as a factor for all positive integers n. in VS .NET
1.30. Prove that xn yn has x y as a factor for all positive integers n. Reading QR Code 2d Barcode In Visual Studio .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Making Quick Response Code In .NET Framework Using Barcode drawer for VS .NET Control to generate, create QR Code ISO/IEC18004 image in Visual Studio .NET applications. The statement is true for n 1 since x1 y1 x y. Assume the statement true for n k, i.e., assume that xk yk has x y as a factor. x Decode QR Code In VS .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET framework applications. Drawing Barcode In .NET Using Barcode encoder for .NET Control to generate, create barcode image in .NET framework applications. Consider
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UPCA Supplement 5 Creation In VS .NET Using Barcode printer for .NET framework Control to generate, create Universal Product Code version A image in .NET applications. Barcode Generation In VS .NET Using Barcode maker for VS .NET Control to generate, create barcode image in .NET framework applications. x x y y xk yk The rst term on the right has x y as a factor, and the second term on the right also has x y as a factor because of the above assumption. Thus xk 1 yk 1 has x y as a factor if xk yk does. Then since x1 y1 has x y as factor, it follows that x2 y2 has x y as a factor, x3 y3 has x y as a factor, etc. Drawing 1D Barcode In VS .NET Using Barcode generator for VS .NET Control to generate, create 1D image in .NET framework applications. MSI Plessey Drawer In .NET Using Barcode creator for .NET framework Control to generate, create MSI Plessey image in .NET framework applications. NUMBERS
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Generating Barcode In ObjectiveC Using Barcode creator for iPhone Control to generate, create barcode image in iPhone applications. Draw Code 128 In None Using Barcode printer for Software Control to generate, create Code 128 Code Set A image in Software applications. The statement is true for n 2 since 1 x 2 1 2x x2 > 1 2x. Assume the statement true for n k, i.e., 1 x k > 1 kx. Multiply both sides by 1 x (which is positive since x > 1). Then we have 1 x k 1 > 1 x 1 kx 1 k 1 x kx2 > 1 k 1 x Thus the statement is true for n k 1 if it is true for n k. But since the statement is true for n 2, it must be true for n 2 1 3; . . . and is thus true for all integers greater than or equal to 2. Note that the result is not true for n 1. However, the modi ed result 1 x n A 1 nx is true for n 1; 2; 3; . . . . EAN128 Decoder In C# Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET framework applications. Recognizing Code 128A In C# Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET applications. MISCELLANEOUS PROBLEMS 1.32. Prove that every positive integer P can be expressed uniquely in the form P a0 2n a1 2n 1 a2 2n 2 an where the a s are 0 s or 1 s. Dividing P by 2, we have P=2 a0 2n 1 a1 2n 2 an 1 an =2. Then an is the remainder, 0 or 1, obtained when P is divided by 2 and is unique. Let P1 be the integer part of P=2. Then P1 a0 2n 1 a1 2n 2 an 1 . Dividing P1 by 2 we see that an 1 is the remainder, 0 or 1, obtained when P1 is divided by 2 and is unique. By continuing in this manner, all the a s can be determined as 0 s or 1 s and are unique. 1.33. Express the number 23 in the form of Problem 1.32.
The determination of the coe cients can be arranged as follows: 2 23 2 11 2 5 2 2 2 1 0 Remainder Remainder Remainder Remainder
1 1 1 0 Remainder 1
The coe cients are 1 0 1 1 1. Check: 23 1 24 0 23 1 22 1 2 1. The number 10111 is said to represent 23 in the scale of two or binary scale. 1.34. Dedekind de ned a cut, section, or partition in the rational number system as a separation of all rational numbers into two classes or sets called L (the lefthand class) and R (the righthand class) having the following properties: I. II. III. The classes are nonempty (i.e. at least one number belongs to each class). Every rational number is in one class or the other. Every number in L is less than every number in R. Prove each of the following statements: (a) There cannot be a largest number in L and a smallest number in R. (b) It is possible for L to have a largest number and for R to have no smallest number. type of number does the cut de ne in this case What (c) It is possible for L to have no largest number and for R to have a smallest number. What type of number does the cut de ne in this case CHAP. 1]

