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Problem 2. Find (with proof) all positive integers n that have an odd number of positive...
Find all integers x, and odd integers n such that, 1 + n^2 = x^3. Please...
Find all integers x, and odd integers n such that, 1 + n^2 = x^3. Please give an explanation with proof thank you.
DEFINITION: For a positive integer n, τ(n) is the number of positive divisors of n and...
DEFINITION: For a positive integer n, τ(n) is the number of
positive divisors of n and σ(n) is the sum of those divisors.
4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
Exercise 17.8 Find all positive integers n for which o(n) = 60. Whenever a mathematician asks...
Exercise 17.8 Find all positive integers n for which o(n) = 60. Whenever a mathematician asks you to find all numbers with certain properties, you are being asked to prove that the values you do find are the only values that work. Hint: Compare the factors of o(n) of the form 1+p+ ... + pk with the positive divisors of 60.
Question 3 (a) Write down the prime factorization of 10!. (b) Find the number of positive...
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).
A perfect number is a positive integer that equals the sum of all of its divisors...
A perfect number is a positive integer that equals the sum of all of its divisors (including the divisor 1 but excluding the number itself). For example 6, 28 and 496 are perfect numbers because 6=1+2+3 28 1 + 2 + 4 + 7 + 14 496 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 Write a program to read a positive integer value, N, and find the smallest perfect number...
4) Let D be the set of all finite subsets of positive integers. Define a function...
4) Let D be the set of all finite subsets of positive integers. Define a function (:2 - D as follows: For each positive integer n, f(n) =the set of positive divisors of n. Find the following f (1), f(17) and f(18). Is f one-to-one? Prove or give a counterexample.
C Programming Write a C program that asks the user for a positive number n (n...
C Programming Write a C program that asks the user for a positive number n (n ≥ 1) then, the program displays all the perfect numbers up to (including) n. A number n is said to be a perfect number if the sum of all its positive divisors (excluding itself) equals n. For example, the divisors of 6 (excluding 6 itself) are 1,2 and 3, and their sum equals 6, thus, 6 is a perfect number. Display the numbers with...
I got a C++ problem. Let n be a positive integer and let S(n) denote the...
I got a C++ problem.
Let n be a positive integer and let S(n) denote the number of divisors of n. For example, S(1)- 1, S(4)-3, S(6)-4 A positive integer p is called antiprime if S(n)くS(p) for all positive n 〈P. In other words, an antiprime is a number that has a larger number of divisors than any number smaller than itself. Given a positive integer b, your program should output the largest antiprime that is less than or equal...
please answer questions #7-13 7. Use a direct proof to show every odd integer is the...
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...
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...
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.
DEFINITION: For a positive integer n, τ(n) is the number of
positive divisors of n and σ(n) is the sum of those divisors.
4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
Exercise 17.8 Find all positive integers n for which o(n) = 60. Whenever a mathematician asks you to find all numbers with certain properties, you are being asked to prove that the values you do find are the only values that work. Hint: Compare the factors of o(n) of the form 1+p+ ... + pk with the positive divisors of 60.
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).
A perfect number is a positive integer that equals the sum of all of its divisors (including the divisor 1 but excluding the number itself). For example 6, 28 and 496 are perfect numbers because 6=1+2+3 28 1 + 2 + 4 + 7 + 14 496 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 Write a program to read a positive integer value, N, and find the smallest perfect number...
4) Let D be the set of all finite subsets of positive integers. Define a function (:2 - D as follows: For each positive integer n, f(n) =the set of positive divisors of n. Find the following f (1), f(17) and f(18). Is f one-to-one? Prove or give a counterexample.
I got a C++ problem.
Let n be a positive integer and let S(n) denote the number of divisors of n. For example, S(1)- 1, S(4)-3, S(6)-4 A positive integer p is called antiprime if S(n)くS(p) for all positive n 〈P. In other words, an antiprime is a number that has a larger number of divisors than any number smaller than itself. Given a positive integer b, your program should output the largest antiprime that is less than or equal...
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...
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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