Prove that N × N × N is countable. Does your argument generalize to the Cartesian product of k copies of N, where k is a positive integer?

Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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Q: 2. Prove that 11" – 7" is divisible by 4 for any positive integer п.

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Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: n^2 +1 is not divisible by 3 for every integer n.

A: Any integer n can be written in the form of: n=3a+b, where b = 0 or 1 or 2 Case 1: If b=0, n2+1=…

Q: Show that for any integer n, n² – 2 is not divisible by 3.

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: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

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Q: Prove that for any integer n, n6k - 1 is divisible by 7 if gcd(n, 7) = 1 and k is a positive…

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Q: Prove that N× N × N× N is countable.

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Q: Prove, by induction, that n + 3n2 + 2n is divisible by 6, for n 2 0.

A: In this problem I have use mathematical induction method two times please read step by step.

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

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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: Prove that for any integer n, we have n - 2 is not divisible by 4.

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Q: Show that for any integer n , n2 – 2 is not divisible by 3.

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Q: Use the factorization theorem to prove that x−c is a factor of x^n−c^nfor any positive integer n.

A: Let's understand the factor theorem before we get into solving the question directly. (x - a) is a…

Q: Prove that if for some integers a, b, c we have that a° + 6° + c° is divisible by 9, thên at least…

A: Here we have to prove that if for some integers a,b,c we have that a3+b3+c3 is divisible by 9,then…

Q: Prove that if for some integers a, b, c we have that a + b3 + c* is divisible by 9, then at least…

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Q: Prove that for any n E N, nº + 2n is divisible by 3.

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Q: Prove that n2-n is divisible by 2 for every positive integer n

A: We have to prove that,  n2-n is divisible by 2 for every positive integer n. We will use…

Q: Theorem 2.9. Let n be a positive integer and write it as n = N²m, where m is square-free. If m…

A: n be a postive Integer and wollte it as mn = N ^ 2 * m where m is square furee. fm Contains no…

Q: Prove that for all positive integers n, 4n-1 is divisible by 3.

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

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Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: Prove by mathematical induction that for all integers n 2 0: a 5" + 2 x 11" is divisible by 3, b 33n…

A: (a) Case 1 : n=1 51+2×111 = 5+2×11= 5+22= 27 As 27 is a multiple of 3 , hence the result holds for…

Q: (d) Prove by induction that 5 "– 1 is divisible by 5 for all n e Z +.

A: we have to show that 5n-1 is divisible by 5 for all n belongs to Z+ by induction for n =1.

Q: (b) Every integer n> I is the product of a square-free integer and a perfect square. [Hint: If n =…

A: By the fundamental theorem of arithmetic for an integer n>1 n=p1k1·p2k2.....prkr where each pi,…

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 there is no integer n for such that 5n+3 is divisible by 5.

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Q: Prove that for any integer a , one of the integers a,a+2,a+4 is divisible by 3

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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: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: Prove that2.7n + 3.5n – 5 is divisible by 24, for all n ∈ N.

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Q: If n = 4k + 3, does 8 divide n2 - 1? (Assume that n and k represent integers.) O Yes, because n2 - 1…

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Q: Prove that n3+(n+1)3+(n+2)3 is divisible by 9 for all non-negative integers  n.

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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: Prove that n2+n  is  divisible by 2, where nN.

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Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

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Q: Prove the given statement: If the product of two integers is not divisible by an integer n, then…

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Q: Prove that n2-n is divisible by 2 for every positive integer n.

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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: (a) Prove that 82n – 12" is divisible by 13 for all nɛ N.

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

Q: P

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Q: Prove, using mathematical induction, that 8n−3n is divisible by 5 for all integers n≥0.

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Q: Show that any integer of the form 6k + 5 is also of the form 3j + 2, but not conversely.

A: 2. We can write 6k+5 = 6k+3+2=3(2k+1)+2 lets write 2k+1=j hence 6k+5=3j+2

Q: 5. Prove that for any integer n, at least one of the integers n, n+2, n +4 is divisible by 3.

A: According to guidelines we can solved only ist question thank you   we have to Prove that for any…

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

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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 n(n + 1)(n + 2) is always divisible by 6

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