Show that the cube for any integer is of the form 7k, or 7k ± 1

Q: Prove that if n is an odd integer, then n3 +n+1 is also an odd integer.

A:

Q: If k is an integer: exp[(i)(pi)(2k+1)] = ?

A: Given that: exp[(i)(pi)(2k+1)]

Q: Prove the square of any integer has the form 4k or 4k+1 for some integer k

A: Consider the set of all integers , To prove that the square of any integer is of the form 4k or 4k+1…

Q: show that if x is an integer then x2+x-41= 0 produce prime numbers

A: Here show that if x is an integer then  produce prime numbers.

Q: Find the solution to 34x = 77(mod 89) that is positive and less than 89.

A: given that                                          34x≡77 (mod 89) we have to find positive…

Q: Let e be a positive real number. Prove or disprove that n E N(n²+e).

A:

Q: The square of an even integer is congruent to 0 (mod 4). Select one: O True O False

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that for a positive integer n we have that  33n+3 - 26n - 27 is divisible by 169.

A: To show  33n+3 - 26n - 27 is divisible by 169.

Q: Suppose that [E:Q] = 2. Show that there is an integer d such that E = Q(square root of d) where d is…

A:

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169.

A: Topic - principle of mathematical induction

Q: Prove that if n is a positive integer, then 133 divides 11"+1 + 122n-1

A:

Q: Prove that 2" >n For all nuatural numbers

A:

Q: Question 6. Let n, m, k be nonnegative integers such that m< k< n. Show that OC) -OG) k k – m

A: To prove: nkkm=nmn-mk-m We know that nk=n!k!n-k!

Q: For every positive integer a, find a composite number n such that n|a^n - a.

A:

Q: QUESTION 12 Show by contraposition that for all integers n, if „2 is odd, then n + 1 is even. For…

A:

Q: Before arriving at his famous conjecture, Goldbach had conjectured that every odd positive integer…

A: Goldbach conjecture says that every integer greater than 5 is the sum of three primes.

Q: If a is any integer, is a(a + 1) even or odd?

A:

Q: Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

Q: QUESTION 9 Use the contrapositive method to show that for any integers a and b, a+b≥13 implies that…

A: In this question, we have to prove that for any integers a and b,  a+b ≥ 13 implies that a≥7 or b≥7…

Q: Give a direct proof of the following statement. If x is an even integer, then x² – 6x + 5 is odd. |

A: Remark: We know that addition or subtraction of an even and an odd integer is always an odd integer.…

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

A:

Q: Show that given any particular real number, such as I, there is NO next real number immediately…

A:

Q: Prove that the fourth power of any integer has one of the forms 5k or5k + 1, where k is an integer.

A: We need to prove that any integer has one of the forms 5k or 5k + 1, where k is an integer.Let p be…

Q: Suppose that all the entries in A are integers and det A = 1. Explain why all the entries in A–1 are…

A: Inverse of a matrix A is calculated by,

Q: Number theory

A:

Q: prove that if a and b are both odd integers, then ab+1 is an even integer

A:

Q: prove that Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…

Q: show that no integer of the form n^3+1 is a prime other than 2

A:

Q: If n is a perfect number, prove Σ d|n 1 2.

A:

Q: Let m and n be positive integers.

A: Given, m and n be positive integers.  To prove that mℤ⊕nℤ is an ideal of ℤ⊕ℤ and that…

Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

A:

Q: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: Prove 8 divides 34h+1+52h+1 whenever h is a positive integer

A: We need to prove that 8 divides 34h+1+52h+1 whenever h is a positive integer by Mathematical…

Q: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: Given: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even

Q: Prove that if m and n are even integers, then mn+ 1-m-n is an odd integer.

A: Given :- m and n are two even integers. To prove :- mn + 1 -m -n is odd integer

Q: If n is an even integer, then 3n° is an even integer.

A:

Q: Assuming every integer is either even or odd, How would I show that for any integer, the difference…

A: Assume that every integer is either even or odd. Let the integer considered say x is even. Then by…

Q: Problem 1. Does there exist a natural number n such that 13 (n² + 1)? Prove your assertion.

A:

Q: For any natural number n, n²+1 is always odd. True False

A:

Q: Prove that for a positive integer n we have that Bn+3 - 26n – 27 is divisible by 169.

A:

Q: Suppose that k and n > k are fixed positive integer. Justify the identity 'n +1° k +1 k j=k

A: We have to solve given problem:

Q: 1. Ifx is an even integer, then x is even.

A:

Q: If n is an odd integer, show that n4 + 4n2 + 11 is of the form 16k.

A: Solution: The objective is to show that given equation is of the form 16k

Q: Ф(тk) — ф(0) — тk

A:

Q: Prove that ei + 01 is odd.

A: Given: e1, e2 are even integers and o1, o2 are odd integers. To prove: e1+o1 is odd integer.