1. Show that if seven integers are selected from the first 10 positive integers, there must be at least two pairs of these integers with the sum 11.              a.) Is the conclusion still true if six integers are selected rather than seven?

Q: Let A be the set of five-digit numbers where only the digits 1, 2 and 3 are included. For example…

A: Given that A is the set of five-digit numbers where only the digits 1, 2 and 3 are included.  So,…

Q: The diagram below shows ten circles joined by five lines with three circles Place all the…

A: Requirements: Numbers should range from 1 to 10.   Every line has three circles.   3. The amount for…

Q: Is the set of all of the integers between 12 and 99 finite or infinite?

A: Given : The set of all of the integers between 12 and 99

Q: Let $n$ be a positive integer. Among C(2n,0),C(2n,1),..., C(2n,2n), C(2n,n) is the largest. True…

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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: 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: Let A be the set of positive integers that are less than or equal to 40 and are multiples of 4 and…

A: Given: Let A be the set of positive integers that are less than or equal to 40 and are multiples of…

Q: Assume n and m are two integers such that n > m > 1. If a bit string of length n+m is randomly…

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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: (a) Show that if seven integers are selected from the first 10 positive integers, there must be at…

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

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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: Suppose that in a particular Finite course, 5/14 of students plan on majoring in math, 11/42 plan on…

A: Given information- We have given the proportion of particular Finite course. P (Major in math) =…

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

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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: 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: 4. Show that if n is a positive integer, then Σ {a₁,...,ak}{1,2,...,n} (Here the sum is over all…

A: Given, If n is a positive integer, then we have to show that ∑a1,.....ak⊆1,2,....,n 1a1a2....ak=n

Q: Discrete

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

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

Q: Show that if there are 8 people in a room, then at least two of them must have been born on the same…

A: There are 7 days in a week.  So we can have 7 people who can have different birthdays from Sunday to…

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

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Q: 2. Use a Venn diagram to illustrate the subset of odd integers in the set of all positive integers…

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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: Show that among any 4 n−1 people, there are either some n of them who mutually know each other, or…

A: Pigeonhole Principle  :  Simple form Theorem 1  :  If n + 1 objects are put into n boxes, then at…

Q: If n > 1 is an integer not of the form 6k + 3, prove that n + 2" is composite. [Hint: Show that…

A: Congruence modulo relation on integers: Let a, b∈Z and let m be a fixed positive integer. Then we…

Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: Given 51 distinct positive integers Strictly of Prove two trom to Sum 99 less than 100,

A: Please find the answer in next step

Q: Suppose there is a currency which only uses 3-cent coins and 7-cent coins. Prove that for all…

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Q: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

A: According to the given question, if any positive integer n belongs to the required subset then it…

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

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

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Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Let S be the set of all (nonnegative) 4-digit integers. How many integers does S contain?

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Q: 6. (a) Show, using the pigeonhole principle, that in any set of 5 integers, at least two have the…

A: (a) Show that any set of 5 integers at least have two the same remainder when it is divided by 4…

Q: Prove that a selection of 7 numbers from the set {1, 2, . . . , 11} contains numbers m and n such…

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Q: Prove that out of 14 people either there are three, each pair of whom are acquainted, or there are…

A: Suppose there is a person A among the 14 people. Look at the following representation, where the…

Q: sum of two even integers is even

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

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Q: How many landline telephone numbers are there with 7 digits such that: (c) the two initial digits…

A: Given: Number of digits in a telephone number = 7 Total number of digits = 10 (0 to 9) Number of odd…

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: How many 5-tuples (x1, x2, x3, x4, X5) of positive integers from 1 to n are there? How many 5-tuples…

A: Since you have asked multiple questions, we will solve the Question Q1 (a) and (b) and (c) for you.…

Q: Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.

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Q: Suppose that 150 people celebrated their birthday in three months, where 65 people celebrated their…

A: Hello! As you have posted more than 3 sub parts, we are answering the first 3 sub-parts.  In case…

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

A: As we know that any truples h,i,j,k,m with n≥h≥i≥j≥k≥m≥1  can be represented as a string of n-1…

Q: Let A be the set of five-digit numbers where only the digits 1, 2 and 3 are included. For example…

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Q: A restaurant sell features a burrito bar in which a customergets to choose one of 2 different kinds…

A: As per bartleby guidelines only one question is answered please upload the other one separately…