9. Let 7 be a prime in Z[i]. (a) Nπ is prime in Z. (b) The number of elements in Z[i], is Nä.

Q: 5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: 5.2.6 Prove by induction that, for all n E N, 1 1.2+2.3+3.4+ · .+n(n+1) = ;n(n+1)(n+2) ...

A: Given statement is 1.2+2.3+3.4+...+n(n+1)=13n(n+1)(n+2). Induction proof is done in following steps.…

Q: 86. Show that Z, = r(cose + isine) and Z2 = r[cos(-6) + isin(-6)] are conjugates. %3!

A: Given,

Q: 13. Floor and ceiling notation were defined in Section 1.1. (a) Show that []+[=n for all positive…

A: Floor and ceiling.

Q: 3. If π is a prime in Z[i], then the number of elements in Z[i], is N(π). 4. If p, q are positive…

A: Let's address each statement one by one:3. If π is a prime in Z[i], then the number of elements in…

Q: (c) Prove that m is prime if and only if y(m) = m - 1.

A:

Q: 6. Prove that m is prime if and only if y(m) = m - 1.

A:

Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: 1 1 1 17. II П for every integer %3D 2i +1 2i+2 (2n +2)!’ n20.

A:

Q: Determine, without solving the equation, the interval where the solution is guaranteed to exist.…

A:

Q: Find the solution of the following differential equations: a) x" + w?x = 8 (t-), x(0) = 0, x'(0) = 1…

A: The given differential equations: Where, Here, the limit does not above 1. Therefore, f(t)=1

Q: 6. If |G| = pq for distinct primes p and q, show that G is not simple. (Note the slight difference…

A: