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

Q: B. Find the value of k that makes x2 + (k + 5)x + (5k + 1) a perfect square.

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Q: ove the statements: e difference of the squares of any two consecutive integers is

A: Introduction: Odd numbers are defined as numbers that cannot be divided into two equal parts. Odd…

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Q: Prove that if n is odd, then 2 is not a square in GF(5n). In other words, prove that there is no…

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A: This is a classic application of pells equation. We first transform the equation into a pells…

Q: a

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Q: exp(ix())= 4k+1 2

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Q: An integer is square-free if it is a product of distinct primes. Show that for each integer k≥ 1…

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Q: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

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Q: Prove that there are no integer solutions to the equation x² = 4y + 3

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Q: 121k²6 = (

A: Solution

Q: 20. (a) Fill in the blank: 1+3+5+..+ (2n – 1) = k=1 (b) Use part (a) to prove that the sum of the…

A: This is an arithmetic series with common difference (3-1)=2 And this has n terms in it So we use the…

Q: Prove that there are 1000 consecutive positive integers that are not perfect squares.

A: Here I give an example. That proves the hypothesis.

Q: Suppose a, b are relatively prime integers. (a) Show that gcd(8a, 8b) = 8. (b) Show that d = gcd(a +…

A: We will use the fact that if a,b are relatively prime integers, then : gcd(a, b) = 1

Q: If the integer K is greater than or equal to 1 and is odd. Prove that the sum of the k powers of…

A: The N must be a perfect square or double square in order for the total of the k powers of positive…

Q: Prove that if n is odd, then 2 is not a square in GF(5n) In other words, prove that there is no…

A: PROOF :

Q: Prove the following statement: There are integers m and n such that m² + n² is a perfect square and…

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

A: Given that 43n+14 We have to prove that there do not exist three consecutive integer values of n for…

Q: Prove: “The sum of two even integers is even.”

A: To prove: The sum of two even integers is even. n is an even number if it can be expressed as n =…

Q: 3 Show thet x+3x+2 has four in 76 zenos.

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Q: Prove that if n is odd, then 3 is not a square in GF(5n).  In other words, prove that there is no…

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Q: Prove that if a and b are both odd integers, then 8 (2a2+36²).

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Q: Verify that 1225 and 41,616 are simultaneously square and triangular numbers. [Hint: Finding an…

A: To check that the given numbers 1225 and 41616 are both squares as well as triangular numbers.

Q: termine k such kx2  _ 4x + 1/2 + 0 that has real and different solution

A: To determine the value of k such that the below equation has real and different solutions.…

Q: 9) The dimensions of a gift box are consecutive positive integers such that the height is the least…

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Q: Denote by an the nth positive integer that is not a perfect square. For example, we have…

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

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Q: Prove that every perfect square is either a multiple of 5 or of the form 5j - 1 or 5j + 1.

A: Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap −…

Q: Show directly that for p a prime integer, p can be written as p=(a)^2 - d(b)^2 then d is a perfect…

A: From the given statement, p be the prime factor and p can be written as p = a2 – db2.To show: d is a…

Q: Prove that there are no integer solutions to the equation x2 = 4y + 3

A: Given equation:- x²=4y+3  We have to prove that the given equation don't satisfy integer pair of x…

Q: Give a direct proof that if m and n are even are both perfect squares, then mn is also a perfect…

A: Given that, m and n are even and both perfect squares.