91Ó°ÊÓ

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) \(-56,-1199\)

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 \(\mathbf{Z}_{n}, n>1\).

A band of 17 pirates captures a treasure chest full of (identical) gold coins. When the coins are divided up into equal numbers, three coins remain. One pirate accuses the distributor of miscounting and kills him in a duel. As a result, the second time the coins are distributed, in equal numbers, among the 16 surviving pirates, there are 10 coins remaining. An argument erupts and leads to gun play, resulting in the demise of another pirate. Now when the coins are divided up, in 15 equal piles, there are no remaining coins. What is the smallest number of coins that could have been in the chest?

Find a simultaneous solution for the system of four congruences: $$ \begin{aligned} x & \equiv 1(\bmod 2) \\ x & \equiv 2(\bmod 3) \\ x & \equiv 3(\bmod 5) \\ x & \equiv 5(\bmod 7) \end{aligned} $$

How many units and how many (proper) zero divisors are there in (a) \(\mathbf{Z}_{17}\) ? (b) \(\mathbf{Z}_{117} ?\) (c) \(\mathbf{Z}_{1117}\) ?

Prove that in any list of \(n\) consecutive integers, one of the integers is divisible by \(n\).

Determine the last digit in \(3^{55}\).

Solve the following linear congruences for \(x\). a) \(3 x \equiv 7(\bmod 31)\) b) \(5 x \equiv 8(\bmod 37)\) c) \(6 x \equiv 97(\bmod 125)\)