# Provide Oz-Yong an explanation of why there are 1,432 soldiers. Use the Chinese remainder theorem on the correct Diophantine equations and show all work.

Math Question
Paper , Order, or Assignment Requirements

Proving set with nth root

A set S with the operation * is an Abelian group if the following five properties are shown to be true:

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Closure property: For all rand t in S, r*t is also in S
Commutative property: For all rand t in S, r*t=t*r
Identity property: There exists an element ein S so that for every s in S, s*e=s
Inverse property: For every sin S, there exists an element x in S so that s*x=e
Associative property: For every q, r, and tin S, q*(r*t)=(q*r)*t

Prove that the set G(the fifth roots of unity) is an Abelian group under the operation * (complex multiplication) by using the definition given above to prove the following are true:
Closure property
Commutative property
Identity property
Inverse property
Associative property

An integral domain Z is a ring for the operations + and * with three additional properties:

The commutative property of *: For any elements xand y in Z, x*y=y*x.
The unity property: There is an element 1 in Zthat is the identity for *, meaning for any z in Z, z*1=z. Also, 1 has to be shown to be different from the identity of +.
The no zero divisors property: For any two elements aand b in Z both different from the identity of +, a*b≠0.
A field F is an integral domain with the additional property that for every element x in F that is not the identity under +, there is an element y in F so that x*y=1 (1 is notation for the unity of an integral domain). The element y is called the multiplicative inverse of x. Another way to explain this property is that multiplicative inverses exist for every nonzero element.

Modular multiplication, [*], is defined in terms of integer multiplication by this rule: [a]m [*] [b]m = [a * b]m

Note: For ease of writing notation, follow the convention of using just plain * to represent both [*] and *. Be aware that one symbol can be used to represent two different operations (modular multiplication versus integer multiplication).

Prove that the ring Z31 (integers mod 31) is an integral domain by using the definitions given above to prove the following are true:
The commutative property of [*]
The unity property
The no zero divisors property

Prove that the integral domain Z31 (integers mod 31) is a field by using the definition given above to prove the existence of a multiplicative inverse for every nonzero element

Abstract Algebra
The solutions are x ≡ 25 (mod 40) and y ≡ 13 (mod 30). Use the Chinese remainder theorem or linear congruences to verify eachsolution, showing all work.

(1) x ≡ 1 (mod 8)

x ≡ 5 (mod 10)

(2) y ≡ 3 (mod 10)

y ≡ 13 (mod 15)

Explain why a solution does not exist for the following systems of congruence’s. Your work should include reduction to a single equation and the greatest common divisor.

(1) z ≡ 1 (mod 8)

z ≡ 4 (mod 10)

(2) w ≡ 3 (mod 10)

w ≡ 9 (mod 15)
Scenario:

Officer Oz oversees several platoons of soldiers. His teenage son, Oz-Yong, would like to know how many soldiers are under his father’s authority. Officer Oz does not know the answer and comes to you with what little information he does possess. He explains that when all soldiers are lined up on land in groups of 23, there are 6 soldiers left over. When all the soldiers jump into boats that hold 17 soldiers each, there are 4 soldiers left over. Finally, when groups of 9 soldiers line up to eat, there is 1 lone soldier who lines up in isolation. Officer Oz is certain that there are fewer than 2,000 soldiers.

Provide Oz-Yong an explanation of why there are 1,432 soldiers. Use the Chinese remainder theorem on the correct Diophantine equations and show all work.