show that no integer of the form n^3+1 is a prime other than 2

Q: TION 9 Prove by Induction that for every natural number n, 3"–1 is a multiple of 2? Attach File…

A: We prove by induction

Q: Determine whether there are infinitely many primes that have a remainder of 12 when divided by 25.…

A: Given question is Determine whether there are infinitely many primes that have a remainder of 12…

Q: Show that there are infinitely many primes.

A:

Q: Prove that the hundreds digit of n20 is even.

A: To Prove that the hundreds digit of n20 is even.

Q: Show that the sum of all  odd  integers between 1 and 1000 which are divisible by 3 is 83667

A:

Q: he sum of any two integers of the from 4k + 1isalways: COdd prime Zero even

A: The sum of two consecutive integers can always be written as 2n+1, but it can be written as 4n+1…

Q: Show that there are infinitely many primes of the form 4k + 3.

A: We have to show that there are  infinitely many primes of the form 4k+3.

Q: rove that the negative of any even integer is even

A: Assume 2n to be an even integer. 2n = even integer -2n = negative of even integer  Let a = -2n be an…

Q: Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder 1, 5 when…

A: Let x be the required number. The number is divisible by 3, so it leaves the remainder 0 when it is…

Q: Find the number of positive integers X smaller than 12849 which have exactly 2022 positive divisors

A: Theory : Total no of divisor of a number, If its prime factorization, given by ∏i=1hpiai Where p is…

Q: Show that the sum of the cubes of the first n positive integers is (n(n + 1)/2)^2

A:

Q: Show that a number with 30 digits cannot have more than 100 prime factors.  

A: Prime factor: A factor of a number which is a prime number. For example: consider a number 26. The…

Q: of the first 150 positive integers?

A:

Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

A:

Q: Find the number of ways a 5-digit positive integer can be made such that in each of them every digit…

A:

Q: Prove: in L number psiz 12, for every prime poiz, [p] = [1], [5) , [7) , or [11]

A:

Q: Show that if any eight positive integers are chosen, two of them will have the same remainder when…

A: Pigeonhole Principle- If n items are assigned to m containers, and m<n,then at least one…

Q: Prove: If x is a positive prime integer, then x+7 is a composite integer. (i.e., not prime)

A: See solution below

Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

A: We know that 2 is even prime. And all other primes are odd.

Q: Show that there are infinitely many primes to prove that there are infinitely many primes of the…

A: This question is related to Number Theory. We solve the result by contradiction.

Q: Prove that 2^10 + 5^12 is a composite number.

A:

Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

A:

Q: a number whose decimal reoresentation repeats is a rational number. true or false

A: A number is rational if and only if its decimal representation is repeating or terminating (i.e.…

Q: How many mombers between 50 and 200 can be expressed as 4k-1, where k is an integer?

A:

Q: How many positive integers less than 10000 are there which contain at least one 2 or at least one 3…

A:

Q: Show that √17 is Irrational number

A:

Q: Find (characterize) all primes p > 3 that satisfy (금) 1.

A:

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

A:

Q: "Every positive integer can be written as the sum of the square of 3 positive integers." prove…

A: Given: "Every positive integer can be written as the sum of the square of 3 positive integers." To…

Q: Show that, selecting 91 positive integers guarantees that there are at least 10 integers with the…

A:

Q: 5. Give a recursive definition of the set of positive integers that are multiples of 5.

A: Solution

Q: Show that there are infinitely many prime numbers.  0 /∈ N. (zero is not an element of natural…

A:

Q: Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5

A:

Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: How many positive odd integers less than 10,000 can be written using the digits 3, 4, 6, 8, and 0?

A:

Q: Show that there are no prime triples other than 3, 5, 7.

A:

Q: Which numbers in Z17 are relatively prime to 17? Enter your answer as a comma separated list of…

A: The objective of the question is determine all al relatively prime to 17 in Z17 .

Q: Prove that 2^(27) - 1 is not a prime number.

A: A prime number is a natural number greater than 1 that is not a product of two smaller natural…

Q: is there integer that has reminder of 2 when its devided by 5?

A: There are infinite number of integers that has reminder of 2 when its divided by 5

Q: There is no integer that has a reciprocal that is an integer.

A: As per the instructions, we will be answering only the first question. For (32), Notice that the…

Q: What is the set of O of odd positive integers less than 10?

A: The odd positive integers less than 10 are 1, 3, 5, 7, 9.

Q: Prove by contradiction that 2020 cannot be written as a sum of 3 odd integers.

A: The objective is to prove  that 2020 cannot be written as a sum of 3 odd integers by the…

Q: Let be a prime number (integer). Show that p2 + 5 cannot be a prime number.

A:

Q: Find all positive integers n such that both n + 2020 and n + 2021 integers. are LVn +1]

A:

Q: Find 2025-3 (mod 212). (Enter the least non-negative integer.)

A: To Find: 2025-3(mod 212) and enter the least non-negative integer.

Q: show that for every positive integer n there is a prime whose decimal expansion contains n…

A: so we have to prove , for every positive integer n there is a prime whose decimal expansion contains…

Q: Give a proof by contradiction that there does not exist an integer which is both even and odd.…

A: Assume an integer x which is both odd and even.

Q: Suppose that n students take a quiz and their scores sum to 150. What is the smallest number n to…

A: Pigeonhole Principle: The pigeonhole principle states that if n items are put into m containers with…