Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

Q: Determine whether the given set is linearly dependent or linearly independent: S =

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Q: ve that the only perfect number of the form n = p²q where p and q are distinct primes is 28.

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Q: There are infinitely many primes of the form 6n+5.

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Q: 6) Apply induction to show that 2+4+ .2n n(n+1), n 21.

A: We prove this by the induction, firstly prove for n=1, then assume it is true for n=k, and then…

Q: Prove that if p and q are distinct primes, then pq | (p9-1+qp-1-1).

A: p and q are distinct prime. We have to show that pq|pq-1+qp-1-1

Q: 1. Prove that nº + n is even for every integer n.

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Q: Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp…

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Q: Prove that every positive integer n>1 has a prime factorization.

A: Prime factorization is a way of expressing a number as a product of its prime factors.

Q: Prove that if a and b are two distinct positive primes, then the number k = ab cannot be abundant.

A: To prove that if ( a ) and ( b ) are two distinct positive primes, then the number ( k = ab ) cannot…

Q: For any integer n ≥ 1, C(6n, 3n) C(6n, 3n-1). O True O False

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Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

Q: .If p 5 is an odd prime, prove that either p? – 1 or p² + 1 is divisible by 10.

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Q: Prove that 3 devides 17k+1-2*5k+3+11k+3whenever k is a non-negative integer.

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Q: Let a and b be natural numbers, where a,b >= 2, such that gcd(a,b)=1 and ab=(x+y)n for some natural…

A: Given that a and b are natural numbers and  a , b ≥ 2 .  Also, given that gcda,b = 1 and ab = x+yn…

Q: Euler's formula for generating amicable pairs of integers is the following rule: = Suppose n> m > 1…

A: Note : In the Euler's formula for generating amicable pairs it is r = 2n+m (2n-m + 1)2 - 1 and not…

Q: Suppose if p and q are odd integers with p!= q (p is not equal to q). Show that there is a unique…

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Q: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

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Q: Prove that n!+1 and (n+1)!+1 integer n. relatively prime for every positive

A: Given, n!+1 and n+1!+1are two numbers and to prove that they are relatively prime.

Q: Find the amount of money owed at the end of 5 years if $60,000 is borrowed at 7% per year compounded…

A: Here we have to find amount of money at the end of 5 years if money borrowed is $60, 000 at 7% rate…

Q: Suppose that a, b are two integers such that 3a? – b3 = 1. Does it follow that a and b are…

A: Given that a, b are two integers such that 3a2-b3=1.

Q: Do all four problems. 1. Prove that there exists a unique prime number of the form n2 – 4, where n…

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Q: 10. Find all primes p such that 29p+1 is a square.

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Q: p, q, pq | (apq − ap − aq  + a) for all a.

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.