Find all integers x, and odd integers n such that, 1 + n^2 = x^3. Please give an explanation with proof thank you.

Find all integers x, and odd integers n such that, 1 + n^2 = x^3. Please...
Problem 2. Find (with proof) all positive integers n that have an odd number of positive divisors (for example 6 has 4 positive divisors 1,2,3,6).
Prove that for all integers n, (-n) mod 2 = n mod 2. Give an example to show that it is not always true that (-n) mod 3 = n mod 3. Professor mentioned to prove for odd and even integers, however, I don't know how to start the proof.
please help me with this question:
19. Give a story proof that 72 +3 for all integers n 2 2. Hint: Consider the middle number in a subset of (1,2.,n +3) of size 5.
19. Give a story proof that 72 +3 for all integers n 2 2. Hint: Consider the middle number in a subset of (1,2.,n +3) of size 5.
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3. Show that if n e Z so that n is odd then 8|n2 + (n + 6)2 +6. 4. (a) Let a, b, and n be integers so that n > 2. Define: а is congruent to b mod n. The notation here is a = b (modn). (b) Is 12 = 4 (mod 2)? Explain. (c) Is 25 = 3 (mod 2)? Explain. (d) Is 27 = 13 (mod8)? Explain. (e) Find 6 integers x...
3- A signal to, is odd if x.(n) = -2.(-n). Show that for such a signal, we have x,0) = 0. Give an example of an odd signal.
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...
3) [3 marks] Use a proof by cases that for all real number x, xs]x]. You may need this definition. For any real numbers x, [x]= x, if x2 0, -x, otherwise. 4) [3 marks] Give a direct proof that If x is an odd integer and y is an even integer, then x + y is odd. 5) [3 marks] Give a proof by contradiction for the proposition in Q4, above. That is, give a proof by contraction for...
Find all integers x, y, 0 < x, y < n, that satisfy each of the following pairs of congruences. If no solutions exist, explain why. (a) x + 5y = 3(mod n), and 4x + y = 1(mod n), for n = 8. (b) 7x + 2y = 3(mod n), and 9x + 4y = 6(mod n), for n=5.
Let n be an odd positive integer. Consider a list of n consecutive integers, not necessarily starting with 1. Show that the average is the middle number (that is the number in the middle of the list when they are arranged in an increasing order). What is the average when n is an even positive integer instead. We learned that for the odd numbers, we would have to show why n-1/2(2k+n)+(k+n) all over n equals k+(n+1)/2.
(2) x + 1 is even -> x^2 is odd (hint: use a direct proof) (a) (0.5 points) What are you assuming is true: (b) (0.5 points) What are you proving is true: (c) (1 point) Complete the proof: