Prove the square of any odd integer is expressible in the form 8n+1 with n ∈Z.

Q: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

Q: Show that the square roots of two successive natural numbers greater than N2 differ by less than…

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

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Q: Prove that the cube of any integer has to be exactly one of these forms: 9k, 9k + 1, or 9k + 8 for…

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Q: 1) Show that the square of an odd integer is odd. 2) Show that the product of two odd integers is…

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

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Q: 5. Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers.

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Q: Provide a proof by contradiction of the statement below. For all integers x, 3x + 2 is not divisible…

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Q: Prove or disprove For every integer m, m(m+1) is even

A: To prove or disprove the given statement- For every integer m , m(m+1) is even .

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

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

Q: Prove that there do not exist three consecutive integer values of nn for which 43n+1443n+14 is a…

A: Given 43n+14 We have to Prove that  ∄  three consecutive integer values of n for which 43n+14 is a…

Q: Use a direct proof to show that every odd integer is a difference of two squares. (For example: 1 =…

A: first we take a odd integer then we write in 2y+1 form then we transfer to difference of two…

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

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

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Q: 11. Prove that if n is a perfect square, then n + 2 is not a perfect square.

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Q: Prove that if n is odd, then 2 is not a square in GF(5n).

A: Given n is odd number We have to show that 2 is not a square in GF5n. We know that the field GF5 is…

Q: Prove that 67n+ 17 cannot be a perfect square for any three consecutive integers n.

A: We will use quadratic residue and proof by contradiction method.

Q: If k is a positive integer, prove the k root of 6 is irrational

A: Given:Given that k is apositive integer.To find:The kth root of 6 is Irrational.

Q: Prove that a positive integer a > 1 is a square if and only if in the canonical form of a all the…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: Let k be some positive integer. Prove that a number of the form 4k +3 can never be the sum of two…

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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: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

Q: Prove that the statement is true for every positive integer n.3 is a factor of n3 + 2n

A: Consider the given statement, "3 is a factor of  n3+2n" for all positive integer n Suppose Pn…

Q: b) Hence or otherwise prove that there do not exist three consecutive integer values of n for which…

A: First let us assume that 29n+5 is a perfect square for some consecutive integers n and n+1. Then…

Q: prove that √2+√3 is irrational

A: For rational we needed form p by q P and q are integer

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: S Prove by mathematical induction that x 1 is a factor of x" 1 for all integers n > 1.

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Q: Prove that the statement is true for every positive integer n.

A: We have, Pn=a2n-1+b2n-1 Let us consider the statement Pn is true for ∀n∈ℕ. Therefore,…

Q: prove the difference of the squares of two z integers is always divisible by 16 (i.e. z1^2 - z2^2 =…

A: We prove that (z_1)^2-(z_2)^2=16k for some integer k, then we say that divisible by 16.

Q: Consider the statement: "there are no integers and such that 20a+4b%3D1." Is this true or false?…

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Q: Prove that 3/3 is irrational.

A: A number which can be written in the ratio , where and are integers are called rational numbers.…

Q: Prove that (3+2/5) is irrational

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Q: Prove that 3, 5, and 7 are the only three consecutive odd integersthat are prime.

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Q: Prove that n2 + 4 is not divisible by 3 for all integers n.

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Q: Prove that 3+√3 is irrational.

A: As per guidelines we are supposed to solve the maximum one questions at  a time kindly post another…

Q: Prove the given statement: The sum of the squares of two odd integers cannot be a perfect square.

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Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: Suppose that [E:Q] = 2. Show that there is an integer d such thatE = Q(√d) where d is not divisible…

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Q: Prove that 2+/3 is irrational

A: Given,          To Prove : 2+3 is irrational.

Q: Use the PMI to prove that 6" – 1 is divisible by 5 for every integer n > 0.

A: To prove that 6^n-1 is divisible by 5 for every integer greater that or equals to 0.

Q: Prove (NOT BY EXAMPLE) that for every nonzero integer, its square is positive.

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Q: Show that Zn has a nonzero nilpotent element if and only if n is divisibleby the square of some…

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Q: A. Prove that n3 + 2n is divisible by 3 for all integers n.

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Q: Prove that if n is an integer, then 2n2 + n + 1 is not divisible by

A: The given problem is to prove the following statement using the possibile cases. We need to find the…

Q: Prove that the square of any odd integer n, (n =4q +3 for some integer q) has the form 8m + 1 for…

A: Here we have to prove that any odd integer n which is of the form (n = 4q + 3) , where q is an…

Q: Prove that if n is an odd integer, then 3n+ 5 is an even integer.

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Q: Prove that the difference of squares of any two odd numbers is múltiple of 8. (SOlve this task…

A: To prove- The difference of square of any two odd numbers is a multiple of 8.   Odd numbers are…