To solve this problem we make use of "combinations of primes" possible to form a factor of given number.
This is also a generalized formula known as "Tau (n)" i.e. number of positive divisors of a given number "n".
1. (10 points) For the following questions, let p, q, r e Z be distinct positive...
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q) is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP + βQ|α, β ∈ Z[x]}. (iii) For which primes p and which integers n ≥ 1 is the polynomial xn − p...
Question 3 (a) Write down the prime factorization of 10!. (b) Find the number of positive integers n such that n|10! and gcd(n, 27.34.7) = 27.3.7. Justify your answer. Question 4 Let m, n E N. Prove that ged(m2, n2) = (gcd(m, n))2. Question 5 Let p and q be consecutive odd primes with p < q. Prove that (p + q) has at least three prime divisors (not necessarily distinct).
6. Let n be any positive integer which n = pq for distinct odd primes p. q for each i, jE{p, q} Let a be an integer with gcd(n, a) 1 which a 1 (modj) Determine r such that a(n) (mod n) and prove your answer.
8) (Problem 17 (a) on page 49) Let p and q be two distinct primes. Show that for any integer a, pq|(a p+q − a p+1 − a q+1 + a 2 ). Hint: Find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo p, and then find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo q. After that, use the following result: Suppose x,...
5. Let P, Q, and R denote distinct propositional variables. Which of the following arguments are valid? Justify your answer. (a) (P+Q), Q+ R), therefore (-PVR). (b) ((PAQ) + R), P, R, therefore Q.
1. Let 21,...,m ER be m distinct real numbers. Define m (p, q)m = p(x;) g(x3), j=1 for all p, q E P = {real polynomials}. Does (-;-)m define an inner product on P? If so, then prove it. If not, then give a counterexample. For which n e N does (-:-)m define an inner product on Pn = {p € P: deg p <n}. Make sure to justify your answer fully!
N=pq with p,q distinct odd primes. Give an expression for the order of (Z/NZ)x in terms of p and q. Then, give an expression for the maximum order of a single element in (Z/NZ)x in terms of p and q.Why does that imply that there does not exist a primitive root modulo N?
8. (a) Prove that if p and q are prime numbers then p2 + pq is not a perfect square. (b) Prove that, for every integer a and every prime p, if p | a then ged(a,pb) = god(a,b). Is the converse of this statement true? Explain why or why not. (c) Prove that, for every non-zero integer n, the sum of all (positive or negative) divisors of n is equal to zero. 9. Let a and b be integers...
7. Let p and q be distinct odd primes. Let a є Z with god(a, M) = 1. Prove that if there exists b E ZM such that b2 a] in Zp, then there are exactly four distinct [r] E Zp such that Zp
B. Let p and q be distinct positive prime numbers. Set a p+ (a) Find a monic polynomial f(x) EQlr of degree 4 such that f(a) 0. (b) Explain why part (a) shows that (Q(a):QS4 (c) Note: In order to be sure that IQ(α) : Q-4, we would need to know that f is irreducible. (Do not attempt it, though). Is it enough to show that f(x) has no rational roots? (d) Show V pg E Q(α). Does it follow...