Question

In: Math

23. Show that for any numbern, n andn+1 are coprime. 28. Show that if (a,b)=1 and...

23. Show that for any numbern, n andn+1 are coprime.

28. Show that if (a,b)=1 and c divides a, then (c,b)=1.

Solutions

Expert Solution

Solution:

Suppose

   and

   is always true

   and      are coprime

  there are integers   and    such that

Also,   

  there is an integer   such that

substitute   in     


Related Solutions

Show that for any square-free integer n > 1, √ n is an irrational number
Show that for any square-free integer n > 1, √ n is an irrational number
Show that, in n-dimensional space, any n + 1 vectors are linearly dependent. HINT: Given n+1...
Show that, in n-dimensional space, any n + 1 vectors are linearly dependent. HINT: Given n+1 vectors, where each vector has n components, write out the equations that determine whether these vectors are linearly dependent or not. Show that these equations constitute a system of n linear homogeneous equations with n + 1 unknowns. What do you know about the possible solutions to such a system of equations?
1. a. Show that for any y ∈ Rn, show that yyT is positive semidefinite. b....
1. a. Show that for any y ∈ Rn, show that yyT is positive semidefinite. b. Let X be a random vector in Rn with covariance matrix Σ = E[(X − E[X])(X − E[X])T]. Show that Σ is positive semidefinite. 2. Let X and Y be real independent random variables with PDFs given by f and g, respectively. Let h be the PDF of the random variable Z = X + Y . a. Derive a general expression for h...
Show that any graph with n vertices and δ(G) ≥ n/2 + 1 has a triangle.
Show that any graph with n vertices and δ(G) ≥ n/2 + 1 has a triangle.
Show that for any k ≥ 2, if n + 1 distinct integers are chosen from...
Show that for any k ≥ 2, if n + 1 distinct integers are chosen from the set [kn] = {1, 2, . . . , kn}, then there will be two integers which differ by at most k − 1. Please demonstrate the steps so that I can learn from it and solve other problems by following the reasoning!
For any r, s ∈ N, show how to order the numbers 1, 2, . ....
For any r, s ∈ N, show how to order the numbers 1, 2, . . . , rs so that the resulting sequence has no increasing subsequence of length > r and no decreasing subsequence of length > s.
Show that (a)Sn=<(1 2),(1 3),……(1 n)>. (b)Sn=<(1 2),(2 3),……(n-1 n)> (c)Sn=<(1 2),(1 2 …… n-1 n)>
Show that (a)Sn=<(1 2),(1 3),……(1 n)>. (b)Sn=<(1 2),(2 3),……(n-1 n)> (c)Sn=<(1 2),(1 2 …… n-1 n)>
(1) Show that the set { 1 m + 1 n : m, n ∈ N}...
(1) Show that the set { 1 m + 1 n : m, n ∈ N} is countable. (2) Show that the set {a + b √ 2 : a, b ∈ Q} is countable. (3) Show that the intersection of two countable sets is countable. (4) Show that the set of all irrational numbers is uncountable. (5) Let C = {0, 1, 2, . . . , 9}. Show that the set C ×C × · · · is...
Music Group Control Group 30 28 30 23 32 25 23 26 28 30 24 20...
Music Group Control Group 30 28 30 23 32 25 23 26 28 30 24 20 26 29 29 20 30 28 26 26 20 27 20 21 24 29 26 23 33 27 21 20 31 26 22 26 26 28 25 27 30 27 SPSS.1 A study reveals that older adults work out a little harder when they listen to music. The table below contains the data from two groups of older adults: one group listened to music...
show that for any n the matrix ring M_n(F) is simple over a field F. show...
show that for any n the matrix ring M_n(F) is simple over a field F. show your work. Do not use quotient rings!
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT