Show that [x + y] ≥ [x] + [y] for all real numbers x and y.
Show by induction that if a and b are integers such that a | b, then a^k | b^k for every positive integer k.
Suppose R is the relation defined on all real numbers by for all real numbers x,y (xRy if |x-yl3) Then for real numbers x and y, xR2y iff
Show that if x and y are real numbers, x2 + y2 >= 2xy and (x + y)2 >= 4xy; When does equality hold (with proof)? Show that if x and y are real numbers, x2 + y2 2xy and (x y) 2Hry. When does equality hold (with proof)?
[9] Given any two real numbers x and y such that x < y, show that there exists a rational number q such that x < a <y.
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...
(10pts) 3. Use direct proof to show that if x and y are positive real numbers, then (x+y)" > " + y".
please answer questions #7-13 7. Use a direct proof to show every odd integer is the difference of two squares. [Hint: Find the difference of squares ofk+1 and k where k is a positive integer. Prove or disprove that the products of two irrational numbers is irrational. Use proof by contraposition to show that ifx ty 22 where x and y are real numbers then x 21ory 21 8. 9. 10. Prove that if n is an integer and 3n...
For all real numbers x and y, if x is not equal to y, x>0,y>0 then (x/y)+(y/x)>2
For any two positive integers a, b, define k(a,b) to be the largest k such that a* | b but ak+1b. Given two positive integers x, y, show that (a) k(a, gcd(x, y)) = min{k(a, x), k(a, y)} for any positive integer a (b) k(a, lcm(z, y)) = max{k(a,a),k(a, y)} for any positive integer a. Hint: Think of the prime factorization of the numbers For any two positive integers a, b, define k(a,b) to be the largest k such that...
QlaJp Is Taise for all res a rec 5. Prove that if a, b, and c are nonzero integers such that alb, bic, and cla, then at least two of a, b, and c are equal. 6. Show that there exist no nonzero real numbers a and b such that Vak + ba 7. Since 2. 1+3.1+5.1 (a) Prove that there do not exist three positive integers a, b, and c such that 2a+36+ Se 11. (b) Use Mathematical Induction...
Show all work. 1. Find two nonnegative real numbers x and y such that x + y = 24 and x2 + y2 is maximized and show why it is a maximum using calculus. 3. The length of a rectangle is decreasing at a rate of 2 cm/sec, while the width wis increasing at the rate of 3 cm/sec. At the moment when the length / is 12 cm and the width wis 5 cm, find the rate at which...