7. Show that for every natural number n2 1, the integer ) + () +*(;) + - +) +22 +... +2" is divisible by 3.

Q: 3. How many 4-digit positive integers are there that are even or contain no O's?

A:

Q: Prove or disprove the validity of the following statement: For any natural number n, n2 + n is even

A: Introduction: To check the validity of a statement is to determine whether the statement is true or…

Q: Prove that for all natural number n, 52n-1 is divisible by 8.

A:

Q: 3. Prove that for each natural number n, 十2 ...

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

A:

Q: What is the converse of: an integer has divisors other than 1 and itself, if an integer is…

A:

Q: 1. Show that subtraction is not associative for the set of integers.

A: Definition :Let "." be a binary operation and A is a setThen "." is said to be associative for A if…

Q: 1. Use a direct proof to show that "If n is an odd number, then n' is an odd number."

A: As per the guidelines, we are supposed to answer only one question at once. Kindly repost the…

Q: 8. How many positive integers less than 1000 a) have exactly three decimal digits? b) have an odd…

A: According to our guideliness we can answer only the first 3 subparts. Positive integers less than…

Q: Show that an integer N is divisible by 9 if and only if the sum of its digits is divisible by 9

A: Let a=anan-1...a3a2a1 be an integer and S be the sum of the digits. Consider a,…

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

A: To prove for any odd positive integer, n, the sum of nconsecutive positive integers is divisible 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: a) How many 3-digit positive integers are there with odd digits?

A:

Q: Compute the sum of all integers between 60 and 150 that are exactly divisible by 8.

A: Consider the given range of number 60 to 150

Q: 1. How many positive integers of 5-digit-number can be formed from the prime numbers starting at 1…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: a. Show that for every non-negative integers m, n, N = 7.5" has exactly (m + 1)(n+1) different…

A: (a) Given that N=7m·5n To find : Number of positive divisors. Please post the multiple questions…

Q: Show that if n is a positive integer then =n(n + 1)2"-2

A: We know the binomial theorem formula isx+yn=∑k=0nCkn xn-kyk, where Ckn is the binomial co-efficient…

Q: Discrete

A:

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

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: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

A:

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

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

A:

Q: 1)How many integers between 1000 and 2000 are divisible by at least one of the integers 3, 5 and 7?

A: Introduction: In mathematics, integers are the sum of positive and negative numbers. Integers are…

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: 27. Determine the number of n-digit numbers with all digits odd, such that 1 and 3 each occur a…

A:

Q: 5. Use the direct proof to show that " If n is an odd integer, then n2 + 3 is even

A:

Q: show that every integer greater than 11 is the sum of two composite integers

A: We need to prove that  every integer greater than 11 is the sum of two composite integers.

Q: 7. What is the second step In proving by mathematical induction that for every positive Integer n,…

A:

Q: Show that if n is an odd integer then []

A:

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

A:

Q: 2. Prove by the method of mathematical induction that for any integer n 21 the expression 8" + 13 is…

A: The solution is given as

Q: Give a recursive definition for the set of all the positive integers that are less than 100 and not…

A: Given: Set of all positive integers that are less than 100. To find: Recursive definition for the…

Q: Negate the statement: For all positive integers, n, there exists a positive integer, k, so that…

A: The negation of a statement is:- Statement:- p implies q. Negation:- p does not imply q.

Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

A:

Q: How many integers between 100 and 999 have distinct digits, and how many of those are even numbers?

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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: 1- The total number of positive integers that can be formed from the digits 1,2,3 and 4 if no digit…

A: NOTE- AS PER POLICY I  HAVE CALCULATED FIRST MAIN QUESTION ONLY. Here Given 4 digits 1,2,3,4

Q: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

Q: 4. Prove that given any integer for n, n3 + 2n will be divisible by 3.

A: We have to prove by induction.

Q: 21. Prove that if n and k are natural numbers such that k < n, then (*:) - (-) - () n k k

A:

Q: Assuming every integer is either even or odd, How would I show that for any integer, the difference…

A: Assume that every integer is either even or odd. Let the integer considered say x is even. Then by…

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

A:

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: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

Q: 5.) Prove that given any integer for n, n³ + 2n will be divisible by 3.

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

Q: Prove: N × N is countable.

A: To prove ℕ ×ℕ is countable, it is enough to show that there is an injection from ℕ ×ℕ to ℕ.