5. A perfect cube is an integer x such that there exists an integer y and x = y³. Prove that the sum of 3 consecutive perfect cubes is divisible by 9. I

Q: Suppose that n pigeons are placed into m holes with m ≥ k(n+ 1) for some positive integer k. Suppose…

A: See the attachment

Q: Let "xy" denote "x does not divide y." Prove by any method. Let a and b be integers. If 3 ab, then…

A:

Q: Prove that the product of any k consecutive positive integers is an integer multiple of k!.

A:

Q: (12) Let p be a prime integer, p≥ 5. Prove that p² - 1 is divisible by 24.

A:

Q: Show that 7 divides 29(52k) + 17(25k-2) for any integer k ≥ 1.

A:

Q: Show that for positive integer n,

A:

Q: n² + n+ 8 2 He has just shown you a convincing proof that if the equation is true for any integer k…

A:

Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

A:

Q: A number n is divisible by 12 if it satisfies ∃k ∈Z(n = 12k) Give a direct proof that, if n is…

A:

Q: Assume an unlimited supply of coins worth 5 cents and 9 cents. Prove that some

A:

Q: 2. Prove that the product of any two odd integers is an odd integer.

A: Let x and y be two odd numbers. If a and b are odd, then a and b can be written as: x=2m+1 , y=2n+1…

Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x > – 1.

A:

Q: prove that for all integers m, n & d if d/m & d/n then d/(2m+3n)

A: We know that if a and b are integers such that a/b then b is a multiple of a. i.e. b=ka where k∈ Z{…

Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

A:

Q: 5. Let r,ne N. Show that the product (n+1)(n+2)...(n+r) of r consecutive positive integers is…

A:

Q: Prove that the summation of three odd integers is odd.

A:

Q: Suppose, and no are the smallest of two positive integers so that 300x is a perfect square and 300y…

A: The given problem is to find the values of x and y and sum of value of x+y for which smallest…

Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: We have to prove that and  are relatively prime for every integer .We know thatThe greatest common…

Q: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

Q: Prove that 3 divides n³ +2n whenever n is a positive integer.

A: Introduction: Consider a starting point for which the assertion is correct. The proposition is true…

Q: 16. What is the greatest integer N such that (10 – N) + (16 - N)> 5?

A:

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: Prove that 6">5" + 9 for any integer n ≥ 2.

A: Mathematical induction is a technique used to prove statements about well-ordered sets of natural…

Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

A:

Q: b) Prove that () - 2) = n², where n is a positive integer.

A:

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

A:

Q: s?

A: *Answer:

Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's Theorem: If p is a prime and a is a positive integer and not divisible by the prime p, then…

Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

Q: 8. Let K be an integer that is greater than 1, a perfect square, and equal to Ei for some integer D.…

A: A perfect square is a positive integer whose square root is a positive integer. Example of perfect…

Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

A:

Q: _4n + 47). Show that there are infinitely many positive integers n that satisfy the division 23|(n²…

A:

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

A:

Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

A:

Q: 5) Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4.…

A:

Q: let M be the greatest integer less than 30 such that M!(M+1)!/2 is a perfect square. Let N be the…

A: Refer to the question we have to find the value of M and N for the given conditions.For solve the…