. Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6.

Q: Prove that any positive integer greater than or equal to 8 can be written as a sum of 3's and 5's.…

A:

Q: 11. Prove that for each integer a, if a² – 1 is even, then 4 divides a² – 1.

A:

Q: Prove that 6 divides n^3 − n whenever n is a non-negative integer

A:

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: Prove that if n is a +ive integer,

A: Given that, M xddxnf(x);p = (-1)npnF(p) We can prove this by mathematical induction.

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

A:

Q: 2. Prove thatn+ 100 > 3" if n is a positive integer with 1 <n< 3. (Hint: Proof by cases)

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Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169. - -

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

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: 5. Prove that the sum of the first n natural numbers is equal to n(n+ 1)/2, and deduce that the sum…

A: Sum of n number is given by

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: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

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: Let n be an integer with n ≥ 2. Suppose that for every prime               p ≤√n, p does not divide…

A: given,                                                  p≤n p doesn't divide n. let n is not a prime…

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

A:

Q: Suppose n and m are integers. Prove by contrapositive: If (n + m) is an even integer and (nm) is an…

A:

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

A:

Q: n any positive integer n, there exist n consecutive composite integers

A: Given that given any positive integer n, show that there exist n consecutive composite integers.

Q: 7) Prove that for any integer n, n' -n is even.

A: We need to prove that for any integer n, n3-n is even, Now, an integer can be either even or odd.…

Q: . Prove that 3| (52n – 1) for every integer n 20.

A: Solution:  To show :   3|(52n-1)     , for every integer n≥0

Q: Prove that a positive integer n is prime number.if no prime p less than or equal to n divides n.

A: Let us assume that the contradiction of the statement , let  n ≥ 0 be a composite number .  Thus , n…

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

A:

Q: Prove the following statement directly from the definitions. The product of any two even integers is…

A: The product of any two even integers is a multiple of 4.

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

A:

Q: 4. Let n and k be two positive integers with n 2. Prove that (n 1)² divides n-1 if and only if n -1…

A: The given problem is to prove the given following statement. Given statement, let n and k be two…

Q: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

Q: Prove that for any integer n if 3n+ 2 is even, then n is also even.

A: As per Bartleby answering guidelines if multiple questions are posted in a single picture then an…

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

A: We need to prove that ,  3n &lt; n!  for every positive integer n greater than 6. We will use method…

Q: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

Q: Prove that 3 divides n3+2n whenever n is a positive integer

A: We use mathematical induction. Let P(n) be "3 divides n3+2n" For n=1 n3+2n=(1)3+2(1)=3 We then note…

Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

Q: Prove that 11 divides 3* 5k + 42k+1 + 33k+2 – 5 * 72k for every k non-negative integer

A: Thanks for the question :)And your upvote will be really appreciable ;)

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: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

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

A:

Q: If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which…

A: The number of r combinations which can be selected from n numbers if repetitions allowed is…

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: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

A:

Q: 1. Show that if seven integers are selected from the first 10 positive integers, there must be at…

A: We can solve the following question by pigeonhole principle

Q: 24. Suppose that I choose six of the first ten positive integers. Prove that I must have chosen two…

A: Rewrite the first ten numbers as follows.

Q: prove that n^2 +1 >= 2^n when is a positive integer with 1<=n<=4

A: To prove n^2 +1 &gt;= 2^n when is a positive integer with 1&lt;=n&lt;=4

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

A:

Q: =) Prove that (*) + ("*) =n“, where n is a positive integer. n", where n is a positive integer. 2

A:

Q: 3. Prove that 0 divides an integer a if and only if a 0.(In your proof. you can freely ILNE your…

A:

Q: is a prime number and that n is an integer such that p|n^2 prove that it follows p|n

A:

Q: Prove or disprove the following: n+1 For any odd integer n, H= ()) 2

A: Let  n=2k+1         -  (1) Where k is an integer. Then n is any odd integer. Now, LHS= n24        -…

Statements and Logic

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