Abstract Algebra

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

(1)  x ≡ 1 (mod 8)

  x ≡ 5 (mod 10)

(2)  y ≡ 3 (mod 10)

  y ≡ 13 (mod 15)

B.  Explain why a solution does not exist for the following systems of congruences. 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.

C.  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.