1. to prove that 11 divides 33n + 10 - 5". Let n E N. Use arithmetic in Z11 (that is, use modular arithmetic)

Q: Prove that N × N × N is countable. Does your argument generalize to the Cartesian product of k…

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Q: 1. Let n be an integer. Prove by contradiction is even.

A: Given that n2+5 is odd. If possible let n is odd. Then we get, n = 2k+1

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: 19. Prove that for each integer a, 3 divides (a + 23a). Hint: sing Theorem 3 0g and tehnia

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Q: prove that is 11^k-4^k is divisible by 7    use mathematical induction

A: we have to show 11k - 4k  is divisible by 7 By using mathematical inductions.

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: 1. Prove that 7" -1 is divisible by 6 for all n > 1.

A: As per Bartleby Guidelines we can solve only one question at a time. Hope you understand

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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

A: Topic:- algebra  Note :- if more than one question uploaded than only first question need to be…

Q: Prove that by mathematically Induction 2n < (n + 1)!, for all integers n ≥ 2.

A: Let s(n):2n&lt;(n+1)!,n≥2...(1) Now, Substitute n=1 in (1), so, 2(2)≤(2+1)!⇒4≤3!⇒4&lt;6 Which is…

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

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Q: Let n be an integer. Prove that if 2n +4 is divisible by 4, then n is even.

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Q: I was asked to prove that 5^(2n+1) + 4^(2n+1) is divisible by 9 for all n that are positive…

A: SOLUTION-

Q: (2) Prove by mathematical induction that for all integers n > 0, the number 42n+1 + (21 · 33"+1) is…

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Q: 1. Show that there are no integers m and n such that VI1. (Hint: Prove by contradiction that we…

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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

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: 24. Show that if p is a prime and k is an integer such that 1sksp-1. then p divides ().

A: 24. we have to show  that for a prime p with r≤k≤p-1      p divides  pk  Notation: pk=p!p-k! k!

Q: 1. Prove that 5 is irrational.

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

Q: prove (1+ )" in <4 for all n21

A: In the given question we have to prove that  1+1nn&lt;4∀n≥1 . we can prove this by two methods one…

Q: By Principle of Mathematical Induction prove 6n 6

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Q: 5. Prove that the integer 53103 + 10353 is divisible by 39, and that 111333 ible by 7.

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Q: 1- Show that if a is an integer, then 3 divides a^3-a. with details plz

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Q: 1. Let n E N. Use arithmetic in Z, (that is, use modular arithmetic) to prove that 7 divides 5 · 6²n…

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Q: Prove that if n is an integer and 5n + 6 is even, then n is even.

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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: 21 divides 4n+1 + 52n-1

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Q: 32. Prove that 3 divides n + 2n whenever n is a positive integer.

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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: 24. Show that if p is a prime and k is an integer such that Isksp-1.then p divides ().

A: See the handwritten solution

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Prove directly from the definitions that for all integers m and n, if m and n have the same parity,…

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Q: Prove that by mathematically Induction 2" < (n + 1)!, for all integers n2 2.

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Q: Let n be an integer. Prove that if 7 divides (n – 2) or 7 divides (n – 5), then 7 divides (n² + 3).

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Q: Prove that for any odd prime p different from 3, ) = 1 if and only if p = ±1 (mod 12). (hint: Use…

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Q: Prove by Mathematical Induction that for any natural number n the number 52n+1 – 22n+1 is divisible…

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Q: Prove that if 10|n, then 10|n^2+n

A: Prove that if 10|n, then 10|n^2+n

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: Show that if p is an odd prime, then −1 is a quadratic residue of p if p ≡ 1 (mod 4), and −1 is not…

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Q: 2. Prove that if m and n are integers and mn is even, then m is even or n is even.

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Q: Prove that 18 divides 2. 192n+1 16 . 174n+1 for all n e N.

A: We need to show that 18 divides 2·192n+1-16·174n+1 Now, a divides b is there exists an integer t…

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

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Q: Prove that 8 divides 9" – 1 whenever n is a natural number. For each natural number n, let P(n) be…

A: To prove that  8 divides 9^n-1 whenever n is a natural

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: 13. |Using the definition of modular arithmetic, prove that if a =b mod n and c = d mod n, then ac =…

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Q: 1. Prove by mathematical induction, for any positive integer n, n³ + 2n is divisible by 3.

A: as per company guidelines we are supposed to answer only one question with maximum three subparts.…

Q: Prove by direct method: for any n integer,if n is odd then 3n+2 is odd

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Q: Prove that If p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

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