Learning Materials
Features
Discover
Chapter 14: Problem 32
Determine the last digit in \(3^{55}\).
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
a) Determine whether each of the following pairs of integers is congruent modulo 8 . i) 62,118 ii) \(-43,-237\) iii) \(-90,230\) b) Determine whether each of the following pairs of integers is congruent modulo \(9 .\) i) 76,243 ii) \(-137,700\) iii) \(-56,-1199\)
Prove that in any list of \(n\) consecutive integers, one of the integers is divisible by \(n\).
a) Find all subrings of \(\mathbf{Z}_{12}, \mathbf{Z}_{18}\), and \(\mathbf{Z}_{24}\) b) Construct the Hasse diagram for each of these collections of subrings, where the partial order arises from set inclusion. Compare these diagrams with those for the set of positive divisors of \(n(n=12 ; 18 ; 24)\), where the partial order now comes from the divisibility relation. c) Find the formula for the number of subringsin \(\mathbf{Z}_{n}, n>1\).
Solve the following linear congruences for \(x\). a) \(3 x \equiv 7(\bmod 31)\) b) \(5 x \equiv 8(\bmod 37)\) c) \(6 x \equiv 97(\bmod 125)\)
Find a simultaneous solution for the system of four congruences: $$ \begin{aligned} x & \equiv 1(\bmod 2) \\ x & \equiv 2(\bmod 3) \\ x & \equiv 3(\bmod 5) \\ x & \equiv 5(\bmod 7) \end{aligned} $$
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine