Compute the sum of all integers between 40 and 100 that are exactly divisible by 3.

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

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Q: Show that = n. mnZ

A: mZ = set of all integers which are divisible by m mnZ = set of all integers which are divisible by…

Q: Show that in a sequence of m integers there exists one or more consecutive terms with a sum…

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

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Q: Suppose that in a class, there are n many boys and n many girls. Among these 2n many people, we will…

A: Let A be a set of n elements. To select r objects from a collection of n elements in the set A…

Q: Compute the sum of the first 60 positive integers that are exactly divisible by 4.

A: The numbers divisible by 4 is:

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: Find S, the sum of first 15 Natural numbers 15

A: We have to find sum of first 15 Natural numbers.   S15? Solution :  As we know that the sum of first…

Q: 3. Show that the weighted arithmetic mean of first n natural numbers whose weights are equal to the…

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Q: Prove the formula for the sum of the first positive integers:

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Q: Use the ertended pigeonhole principle to show that there is at least 15 ways to choose 4 integers…

A: Extended Pigeonhole Principle:      If there are n pigeons and m pigeonholes and n>m. Then, there…

Q: Determine the number of permutations of the set {1 ,2 ,..., 14} in which exactly 7 integers are in…

A: Solution-:

Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: Prove that the summation of three odd integers is odd.

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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>1. All prime divisors of n!+1 must be bigger than n. This is…

Q: How many triples of integers from 2 through 16 can be formed in which the integers are in increasing…

A: Any triple of integers (i, j, k) with 2 ≤ i ≤ j ≤ k ≤ 16 can berepresented as a string of 15 – 1…

Q: Prove that the sum of any five consecutive positive integers is always a multiple of five.

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Q: The sum of all positive integers less than 300 which are divisible by 3.

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

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Q: Determine the identity element of the set N={1,2,3,..} of positive integers with addition.

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Q: Find the sum of the positive even integers up to and including 50.

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Q: Use mathematical induction to prove that any amount of postage that is an integer amount greater…

A: We have to prove that any amount of postage that is an integer amount greater than 53 cents can be…

Q: Prove by Mathematical Induction that 10" - 1 is divisible by 9 for all positive integers n.

A: In proving some statements for all positive integers, we follow the mathematical induction. The…

Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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Q: Use Fermat's Theorem to show that n 13 - n is divisible by 2730 for all n.

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Q: Prove N.又

A: Note as per our guideline we are allowed to do 1 question at a time. Please post other questions…

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 primes of the form 6k + 5.

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Q: The sum of all positive integers less than 300 which are not divisible by 3.

A: Sum of n positive integer is given by, Sn=nn+12

Q: Compute the sum of the first 50 positive integers that are exactly divisible by 5.

A: To determine sum of the first 50 positive integers that are exactly divisible by 5.

Q: How many integers in the set {1,2,3,4, ..., 200} is divisible by at least one of the primes 2, 5 and…

A: Just use inclusion-exclusion principle for three sets.

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: 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: find the sum of all even intergers less than 1000 that are exactly divisible by 7

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Q: Show that the number of partitions of n in which odd parts are not repeated but even parts can occur…

A: Let n=4 Then number of partitions of n=4 in which odd parts are not repeated but even parts can…

Q: Construct four relatively prime integers a, b, c, d such that no three of them are relatively prime.

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Q: Determine the number of ways a computer can randomly generate one or more such integers from 1…

A: There are 5 prime numbers between 1 to 12 . Which are 2 ,3 , 5, 7 , 11.

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

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Q: compute the sum of all integers between 60 and 110 that are exactly divisible by 7

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Q: The sum of all positive integers less than 400 which are divisible by 5.

A: Positive integers less than 400 which are divisible by 5 are⇒n=79 5,10,15,20,...,395 The above…

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

A: We have to prove by induction.

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

A: We can solve this question by pigeonhole principle

Q: 7. How many positive integers between 100 and 999 inclusive are divisible by 7?

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Q: that the set of negative integers divisible by 3 is countable

A: Set of negetivee integers divisible by 3 isS=-3n, n∈ℕf: ℕ→S   defined asf(n) = -3n,To check…

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

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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: 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…