Show that 5n + 3 and 7n + 4 are relatively prime for all n.

Q: Use mathematical induction to prove that (n³ + 3n² + 2n) is divisible by 3 for all n > 1.

A: See the attachment

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 positive integer n: (2n)! VT. 22n . n! Г 2

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Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: 3n+2 and 5n+3 are relatively prime for every integer n.

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: prove via Mathematical Induction that 3n +1 is divisible by 2 for all n>=1

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Q: By Principle of Mathematical Induction prove 5" -1 is divisible by 2 for all n>1

A: To prove: Proof: Use the first principle of mathematical induction that holds for natural numbers.

Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

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Q: Show that n!> n3 if n is large enough.

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Q: Prove that 4 < n! for all n 2 10 by induction

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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: Prove that 3n<n! if n is an integer greater than 6

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Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

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Q: Prove that n! > 2" for n 2 4.

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Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n&gt;1. All prime divisors of n!+1 must be bigger than n. This is…

Q: prove that n(13n²-1) is divisible by 6 for every n ≥1

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Q: If n is a positive integer prove that 3^(2n) - 26n -1 is divisible by 676

A: if n is a postive integer prove that 3^(2n) - 26n -1 is divisible  by 676

Q: 3. Prove that if n is an odd integer then 7n+5 is even.

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Q: The sum of any two integers of the from 4k + 1is always: Even Odd prime Zero Windows b

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Q: Prove by cases that n2−1 is divisible by 3 when n is an integer not divisible by 3.

A: The detailed solution is given below Thank you

Q: Show that if n is a positive integer then =n(n + 1)2"-2

A: We know the binomial theorem formula isx+yn=∑k=0nCkn xn-kyk, where Ckn is the binomial co-efficient…

Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

Q: Prove or disprove: 3n = ω(2^n ).

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Q: Prove by induction that 1) 2 and 3 can divide n^3-n for all n ∈ N 2) n^2-7n+15 >=0 when n>=3 for…

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Q: イの Prove by mathematical induction that 3" + 7 is divisible by 8 for all integers n > 0

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Q: Show that there are infinitely many primes of the form 6k + 5.

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Q: Use the binomial theorem to prove (2n choose k ) x  2^(2n-2k)= (2.5)^2n

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Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

Q: 512|8.5n + 6.32n -80n -14

A: Given: 512|8*5n + 6*32n -80n -14

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: 7. What is the second step In proving by mathematical induction that for every positive Integer n,…

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Q: Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

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Q: Show by mathematical induction, that for all n 2 1 n! > 2n-1

A: To show → by mathematical induction that                       V n ≥ I  , n I ≥ 2n-1Proof →Let ,…

Q: Show that if 2n−1 is prime then n is prime.

A: We have to show that if 2n-1 is prime then n is prime.

Q: Let p and q be odd primes. Prove that (p^2) q is not a perfect number. [Hint: Factor 1 + p + q + pq…

A: Let σp2q=σp2σq Then we have  σp2q=σp2σq=1+p+p21+q=p3-1p-1q2-1q-1≤p3p-1q2q-1

Q: Prove that 11 is the only prime number of the form 4n2 - 25.

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

Q: Show that 5 | n5-n for all integers n.

A: To prove the above statement we will use the principle of mathematical induction which is divided in…

Q: Show that if 2m +1 is an odd prime, then m 2" for some nonnegative integer n.

A: We solve this question by contradiction

Q: Q6 Use mathematical induction to show that n > 2n +1 for all integers n > 2.

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Q: Use mathematical induction to show that 3 divides n3 + 2n whenever n is a nonnegative integer.

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Q: 2. Prove that if 2n² + 7n is even and n is an integer, then n is even.

A: We have to prove that the given statement is true by the help of mathematical induction

Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

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

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Q: Find σ(n) if n = 2^k-1 (2^k - 3) where 2^k - 3 is prime

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Q: Prove that there are infinitely many primes of the form p = 4n + 3.

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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…

A: Table of values for 10n-1 and 10n+1 for n between 1 and 20. n 10n-1 10n+1 1 9 11…

Q: Show that if n is a positive integer then n(n +1)2"-2 k=0

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Q: Use the binomial theorem to show that for all integers n 2 1, 3" = (6) + 2 (4) + 2° (") -

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.