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Let P and Q be two projectors, such that PQ zQP prove that (PQ) is a...
5.104. Let p and q be irreducible elements of a PID R. Prove that R/(pq) = R/(p) x R/(q) if and only if p and q are not associates.
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...
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...
2. Suppose P and Q are positive odd integers such that (PQ)-1. Prove that Qm] Pn] P-1 0-1 0<m<P/2 0<n
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.
Fourth Homework (1) Let P-(**.0) and Q ( . (a) Find the pole of the line PQ (b) Find the parametrization of the line PQ (c) Does (ch,顽週lie on the line PQ? 克,2 7, ) lie on the line PQ? (2) Find the distance between the lines (1,0,-1) + t(2,3,0) and m (2,-1,3) +s(0, 1,2). (3) Let A and B be two distinct points of S2. Show that X e I d(X, A) = d(X, b)) is a line and...
2. (a) Prove that the following sequents cannot be valid: (i) ( PQ) V ~RE (~Q ^ R) P (ii) PQ, R=~SE (PVR) = (Q V S)
plz solve both 6. Let u = PQ and v = PR P= (5,0,0), Q = (4,4,0), R = (2,0,6) a. Find u v b. Find v.v 7. u = 4i + 3j + 6k V = 5i +2j+k a. Find ux v b. Find vxu c. Find vxv
Consider the points P(0,0,9) and Q(-3,3,0). a. Find PQ and state your answer in two forms: (a,b,c) and ai + bj + ck. b. Find the magnitude of PQ. c. Find two unit vectors parallel to PQ.
QUESTIONS Problem 3. Let P, Q be nxn matrices with PQ = QP. Suppose that is nonsinsingular and veR" is a nonzero eigenvector of P. Determine which of the following statements is True. e: v and Qü are eigenvectors of P with the same eigenvalues. 12: v and Qü are eigenvectors of P with distinct eigenvalues. T: Qü is not an eigenvector of P. V: None of the other answers. e оооо