23. Prove that for any integer a, 9/ (a² – 3). -

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A: We have to prove: If k is a positive integer, then 3k+2 and k+1 are relatively prime.

Q: 11. Prove that for each integer a, if a² – 1 is even, then 4 divides a² – 1.

A:

Q: Prove that the analogue of lemma 1.23 is not true for numbers of the form 4n+3 where n is an…

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Q: Prove the following by contrapositive: If x is an even integer, then x² is even.

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Q: Show by contraposition that for all integers n^2, if  is odd, then n + 1 is even.

A:

Q: Prove by mathematical induction that j=1 for all positive integersn. AI

A: To prove:                                                                        ∑j=1nj2≥∑j=1nj2,…

Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

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

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Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

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Q: Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y is even

A: Suppose x and y are odd integers. To show x+5y2 is even. Note: Square of an odd number is odd.…

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

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

Q: 6"- (-4)" %3D k 1 k odd IM- 2]

A:

Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: Prove, for all positive integers n, 1 3 2n – 1 1 - 2 4 2n (3n

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Q: () is even. 3. Prove that for all positive integers n, the number

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Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: 1. Prove that 5 is irrational.

A: Consider the provided expression, 5 Show that the above expression is irrational. We proof by…

Q: -ん 2 kodd

A:

Q: 3. Prove by contradiction, that if x and y are odd integers, then there is no integers z such that…

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Q: - Prove that for any integer a, 9/a² – 3). |

A:

Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

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Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: (5) Suppose thath and k are positive integers and that n = hk. Write a combinatorial proof that (h!)…

A: Suppose h and k are positive integers and that n=hk. Claim: n!h!k is an integer. Proof: Consider the…

Q: 13. Prove by Math Induction e* > 1 + x for all positive integers x > 1

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Q: Prove that 4.999...=5.000...=5

A: Proof:Let x = 4.999, then 10x = 49.99.

Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: Given 51 distinct positive integers Strictly of Prove two trom to Sum 99 less than 100,

A: Please find the answer in next step

Q: Suppose that k and n > k are fixed positive integers. Justify the identity k+1,

A: Given k and n≥k are fixed positive integers.

Q: prove that it ņ is an 3n +2 is odd then integer and is add

A: This is related to Number System

Q: Prove that if ni is a positive integer, a± 0

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Q: prove that IIaind = nd(n)/2.

A: Analytic number theory

Q: Prove that if n, k are positive integers with k | (n +4) and k| (n – 1), then either k = 1 or k = 5.…

A: k | (n+4)  k | (n-1)  => k | {(n+4) - (n-1)} => k | 5

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

A: We use mathematical induction. Let P(n) be "3 divides n3+2n" For n=1 n3+2n=(1)3+2(1)=3 We then note…

Q: Prove that 11 divides 3* 5k + 42k+1 + 33k+2 – 5 * 72k for every k non-negative integer

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

Q: If nis a positive integer 1 then prove (D - a) n!

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Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n V3n

A: For all positive integers n, we need to prove that, 12·34···2n-12n<13n

Q: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

Q: 2. Prove that a positive integer n is even if and only if nis even.

A: We prove that a positive integer n is even if and only if n2 is even.

Q: Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.

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Q: Prove that n² + 1> 2" when n is a positive integer with 1<n<4.

A: Given: is a positive integer with. To prove: .

Q: =) Prove that (*) + ("*) =n“, where n is a positive integer. n", where n is a positive integer. 2

A:

Q: 6. Prove that for n> 2, 2(;) + (*) =n². п

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Q: prove that, it ņ is an 3n+2is oald then integer and nis add

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Q: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…