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

Q: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

Q: Prove that 3"< n! whenever n is a positive integer greater than 6.

A: Here we are to prove 3n&lt;n ! whenever n is a positiveinteger greater than 6.To solve this problem…

Q: Prove that for every non-negative integer n the value of n² + n+41 is prime ?

A: The number n² + n + 41 is not prime for every natural number n. Let, f(n) = n² + n + 41

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

A:

Q: Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contradiction.

A: Given: 3n + 2 is an even integer Let n is not even, thus n is odd. Therefore, by using the…

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 or disprove: For every positive integer n, 22n – 1 is divisible by 3. -

A: I have proved the result using mathematical induction

Q: Prove that the statement is true for every positive integer n.

A: Given that, 2 is a factor of n2+n   We have to prove this statement is true for every positive…

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: Prove there are infinitely many prime numbers

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Q: Prove that the statement is true for every positive integer n.

A: Given arithmetic progression (AP) is: 2+5+8+.........+3n-1 Here the first term is 2 and the common…

Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n.…

Q: Prove that n²– 3n + 1 is odd for every integer n.

A: To prove that n^2-3n+1 is odd for every integer n.

Q: Prove that for all natural numbers n, (n^3/3) + (n^5/5) + (7n/15) is an integer

A:

Q: Prove that there are infinitely many prime numbers

A: Let us assume that there exist finitely many prime numbers (Let n). Here, n is a positive integer.…

Q: Prove or disprove: 2n + 1 is prime for all nonnegative integers n.

A: Prove or disprove: 2n + 1 is prime for all nonnegative integers n.

Q: Let n be a positive integer. Prove that (:) n +2. +...+n. n2"-1.

A:

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: Prove that the statement is true for every positive integer n.

A: we have to prove that the statement is true for every positive integer n. 4 is a factor of 5n-1

Q: Prove that 6 divides n3-n for all non-negative integers n.

A: See the detailed solution below.

Q: Prove that there is some positive integer n such that n, n + 1,n + 2,... , n + 200 are all…

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Q: Prove that the statement is true for every positive integer n.3 is a factor of n3 + 2n

A: Consider the given statement, "3 is a factor of  n3+2n" for all positive integer n Suppose Pn…

Q: Prove that if n is an integer and n² is an even integer, then n is an even integer

A:

Q: Let n be a positive integer. Prove that if n2 is even, so is n.

A: We have,  n be a positive integer and an even number. Therefore, n=2k; k∈ℝ

Q: Show that there are infinitely many integers n such that 4n^2+1 is simultaneously divisible by 13…

A: We need to prove there are infinitely many integers n such that 4n^2+1 issimultaneously divisible by…

Q: Prove that for any integer n, n' -n is even.

A: CASE I When n is even Let n=2m,m∈N⇒n3−n=2m3−2m⇒2m4m2−1=2η,where η=m4m2−1Thus,n3−n is even when n is…

Q: Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

A:

Q: Prove that for every positive integer n, there are n consecutive composite integers.

A: This question is related to Descrete Mathematics

Q: Prove that there exists a positive integer n >1 such that 2n2 + 3n + 11 is prime.

A:

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

A: “Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: Prove or disprove that for every positive integer n, if n is even then n^{2}+1 is prime.

A: False

Q: Prove that 2n>n2 if n is an integer greater than 4.

A: According to the given information, it is required to prove the given statement.

Q: prove that it is not possible to find four distinct prime numbers p, q, r, s such that ps − qr = 0.

A:

Q: 7. Prove that two consecutive integers are always relatively prime. That is, show: gcd(a, a + 1) = 1

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Q: Prove that the statement is true for every positive integer n.

A: Given, Prove this statement is true for every positive integer n: 2+4+6+..+2n=n(n+1)

Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

A:

Q: Prove that n2 +1 > 2? when n is a positive integer with 1<n<4.

A:

Q: Prove that there exists no rational number x such that x2 = p, where p is a prime number (A prime…

A: We need to prove that there exists no rational number x such that , x2 = p  ,  where p is prime…

Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

A:

Q: Prove that the statements here are true for every positive integer n. 1 + 3 + 5 + · · · + ( 2n - 1)…

A: Show that for every positive integer n    1 + 3 + 5 + · · · + ( 2n - 1) = n2

Q: Prove that for every non-negative integer n the value of n^2+n+41 is prime

A:

Q: Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

A:

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 there are infinitely many primes of the form p = 4n + 3.

A:

Q: Prove that n2 − 3n + 1 is odd for every integer n.

A: Prove the result using mathematical induction. P(n):n2-3n+1. Check whether the result is true for n…

Q: Prove by contradiction that 6n + 5 is odd for all integers n.

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Let n be any integer. Using a proof by cases, prove that n4 is either in the form of 5k or 5k+1 for…

A:

Q: Prove that the statement is true for every positive integer n.

A: a&gt;1, then an&gt;1

Q: 1 Prove that if z is a real number such that z+- is an integer, then for all natural numbers 1 n, z"…

A: Given that, if z is a real number such that: z+1z is an integer,Show that for all n,zn+1zn is an…