 7.1: Calculate <f>(lOOl), </>(5040), and </>(36,000).
 7.2: Verify that the equality <f>(n) = <f>(n + 1) = <f>(n + 2) holds whe...
 7.3: Show that the integers m = 3k 568 and n = 3k 638, where k 0, satisf...
 7.4: Establish each of the assertions below:(a) If n is an odd integer, ...
 7.5: Prove that the equation <f>(n) = <f>(n + 2) is satisfied by n = 2(2...
 7.6: Show that there are infinitely many integers n for which <f>(n) is ...
 7.7: Verify the following:(a) For any positive integer n, ,Jn:::; <f>(n)...
 7.8: Prove that if the integer n has r distinct odd prime factors, then ...
 7.9: Prove the following:(a) If n and n + 2 are a pair of twin primes, t...
 7.10: If every prime that divides n also divides m, establish that <f>(nm...
 7.11: (a) If <f>(n) In 1, prove that n is a squarefree integer.[Hint: A...
 7.12: If n = p1 p2 pr, derive the following inequalities:(a) cr(n)<f>(n) ...
 7.13: Assuming that d I n, prove that (d) I (n ).[Hint: Work with the pri...
 7.14: Obtain the following two generalizations of Theorem 7.2:(a) For pos...
 7.15: Prove the following:(a) There are infinitely many integers n for wh...
 7.16: Show that the Goldbach conjecture implies that for each even intege...
 7.17: Given a positive integer k, show the following: (a) There are at mo...
 7.18: Find all solutions of <f>(n) = 16 and (n) = 24.[Hint: If n = p1 p2 ...
 7.19: (a) Prove that the equation (n) = 2p, where pis a prime number and ...
 7.20: If p is a prime and k 2:: 2, show that ((pk)) = pk2</J((p  1)2).
 7.21: Verify that </J(n)a(n) is a perfect square when n = 63457 = 23 31 89.
Solutions for Chapter 7: EULER'S PHIFUNCTION
Full solutions for Elementary Number Theory  7th Edition
ISBN: 9780073383149
Solutions for Chapter 7: EULER'S PHIFUNCTION
Get Full SolutionsChapter 7: EULER'S PHIFUNCTION includes 21 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Number Theory was written by and is associated to the ISBN: 9780073383149. Since 21 problems in chapter 7: EULER'S PHIFUNCTION have been answered, more than 5979 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Number Theory, edition: 7.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.