Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 < n < 20. Circle any numbers that are prime. In your table, is it true that for each n, at least one of the numbers 10n ± 1 is prime? Prove that there are infinitely many values of n for which 10n – 1 and 10n + 1 are both composite.

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

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A: x = 35×k          k is an integery = 33×l           l is an integer

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Q: Give a formula for the nth positive even integer: Here is a theorem: 1+ 2+3 + ... + n п(n+1) n = Use…

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Q: Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for…

A: Either n is a prime or it is composite; in the former case, there is nothing more to prove. If n is…

Q: 9. Show that there are infinitely many primes of the form 16k + 1. More generally, show that for any…

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Q: Given n < 10" for a fixed positive integer n°2, prove that (n + 1)* < 10*1.

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Q: Give an inductive proof that the Fibonacci numbers Fn and Fn+1 are relatively prime for all n ≥ 0.…

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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: For each nonnegative integer i, what is the least residue modulo 9 or 10i?

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Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: Prove 1 +,+ i++ = 2 – for all positive integers n. |

A: We want to prove 1 + 12 + 14 + ... + 12n = 2 - 12n

Q: Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if…

A: Given, r = a/b, where b = 2n5m for some n, m ≥ 0. We know, that any integer divided by 10N has a…

Q: Find all integers n such that (n^2) - 1 is prime.

A: n² - 1 = (n-1)(n+1)  Now, if (n² - 1) needs to be a prime, then, either n-1 is equal to 1 or, n+1 is…

Q: Find a closed formula for a, if ak = 5n² + 4n k=1 an =

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Q: Let n∈Z.  Prove that x is odd if and only if 5x + 4 is odd.  Identify the type of the above…

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Q: Prove that 2n &gt; n for all positive integers n.

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Q: Show that a fraction r = a/ b in lowest terms has a finite decimalexpansion if and only ifb = 2n5m…

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Q: Find all integers n > 1 such that 1n − 1)" is +2+...+(n divisible by n.

A: To find all the integers n&gt;1 such that 1n+2n+...+n-1n is divisible by n

Q: 111111 is prime. Prove that, in any radix b, The number 11111 (11...1), can be prime only if the…

A: As per the question we have to prove that the number 111...1 containing n number of 1's will be…

Q: 10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

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Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: If n > 2 is prime then n | a"-1 1 for all positive integers a with gcd(a,n) = 1. Select one: O True…

A: True

Q: Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for…

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Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

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Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p is not prime in Z[i]

A: Given that p is odd prime. This implies that p is not equal to 2.

Q: (a) Prove that if n is an even integer then (n – 1)² + n is odd. (b) Given that n is an integer,…

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Q: For all integers n>0, prove: 2n < (n+2)!

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Q: R.4. Prove that for every integer n, n³ is even if and only if n is even.

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Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 <n < 20. Circle any numbers that are…

A: Composite numbers contain prime factors other than 1. Such as if a number has prime factors of 2, 3,…

Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: п Find a formula for II (1- ) : and verify. k2 k=2

A: We will expand the terms and check which terms are getting cancel each other

Q: 15. Prove that for each natural number n, s+ (4n-3)(4n+1) 4n+1 ...

A: Given The given expression is 11·5+15·9+·  ·  ·+14n-34n+1=n4n+1

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

A: If n is an an odd integer then we have to show that n4 + 4n2 + 11 is of the form 16k for some…

Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 <n < 20. Circle any numbers that are…

A: We know that if a number k is divisible by m   Then  There exists a natural number n such that k=nm

Q: Let n > 1 be an integer. Show that if pn for all primes p≤ √n, then n is either prime or the product…

A: Introduction: Proof by contradiction, sometimes referred to as indirect proving or the reductio ad…

Q: (5n)! 40 n! is natural number for every natural number n. Prove that

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Q: Let n>=2 and k be any integer, then prove that (n-1)^2 divides (n^k )-1 iff n-1 divides k.

A: See the attachment

Q: Let n be an odd positive integer. Find all perfect numbers of the form nn +1.

A: n=1,3,5,7,9,11,13,...nn+1=11+1,33+1,55+1,77+1,99+1,... Sequence is: 2, 28, 3126, 823544, 387420490,…

Q: Prove that n^2 is an even number if and only if 3n + 1 is an odd number by proving  If n^2 is even…

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Q: Prove that if n = p1a1...prar, where the pi are prime, then σ(n)ϕ(n) = n2(1-p1-a,-1)...(1-pr-a,-1

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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: Prove that for all positive integers n, we have 6″ Σ (1) * _ "(-¹)" 2 k=1 k odd IG

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Q: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

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Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

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Q: There is an integer n such that 7n2 – 5n + 11 is a prime number. -

A: Let fn=7n2-5n+11for n=1f1=7×1-5×1+11      =13 , which is prime number for n=2f2=722-5×2+11…