MAS 4214/001 Elementary Number Theory, Fall 2013
CRN: 80553
Answers, Set 6
Qyestion 1.(2 pts)Find the remainder: 27!mod 899. (Note that 899 = (29)(31).)
Question 2. (1 pt)(Numerical examples for the next problem). Letm≥2, and letUbe the set of integersxbetween 1 andmthat are relatively prime tom. That is,U={x: gcd(x, m) = 1,1≤x≤m}.PartitionUinto two sets:U1={x∈U:x2≡1},andU2=U−U1.That is,U1is the set of integers inUthat are self-inverses modulomandU2is the set of integers inUthatare not self-inverses modulom. LetP, P1, P2, respectively, be the products of the integers inU, U1, U2. Foreach of the following two vlues of m, determine the setsU, U1, U2, and the remainders whenP, P1, P2aredivided bym.(a)m= 10.(b)m= 15.
Solution.