Suppose that n students take a quiz and their scores sum to 150. What is the smallest number n to gurantee that at least two students got the same grade (assume that the quiz score of each student is an integer between 0 and 100)? Hint: Suppose the scores are all different and use Pigeonhole Principle.

Q: Use a direct proof to show that the sum of two odd integers is even.

A: Types of integers : Odd integers : The integer which is not divided by 2 . It has the form 2k+1…

Q: 3) Sarah writes down random positive integers when she gets bored. Prove that if Sarah writes 1001…

A: Given:  Sarah writes 1001 numbers.  Since there are 10 digits, i.e., there are only 103 = 1000…

Q: 2. Prove that " The sum of two odd numbers is even: using direct proof.

A:

Q: Prove that the hundreds digit of n20 is even.

A: To Prove that the hundreds digit of n20 is even.

Q: Suppose that you have 19 balls to place into four different bins. Nine of the balls are red, while…

A: Given data: The total numbers of balls are n=19. The numbers of red balls are r=9. The numbers of…

Q: Suppose that 54% of people own dogs. If you pick two people at random, what is the that they both…

A: Given: P(Dog) = 0.54 The probability that they both own a dog can be calculated as: P(Both own a…

Q: If the first three Fibonacci numbers are given as x, = 1, X2 = 1 and x3 = 2, then what is the least…

A: This is a question of number theory therefore from the subtopic of counting.

Q: Suppose you are at a party with 19 of your closest friends (so including you, there are 20 people…

A:

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: Using the extended pigeonhole principle, show that if there are 30 students in a class, then at…

A:

Q: Find the smallest distinct postive numbers that provide a counterexample to show the statement is…

A: Let the odd numbers be 1 and 3 and the even number be 2, According to the question, The sum of any…

Q: In a set of 2 positive integers, how many ways it can be chosen in less than 50? 1.

A:

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: 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…

A:

Q: If seven integers are chosen from between 1 and 12 inclusive, must at least one of them be odd? Why?

A:

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: 5. Is it possible to plant 10 trees in 10 straight rows, with 3 trees in each row? Hint: Desargues…

A:

Q: 3. Let n be an integer, show that if n is even, then n is also even using indirect proof.

A: Given : Let n be an integer.   To show that if n2 is even, then n is also even.

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

A:

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

A:

Q: Show that for all n:  ceil(n) = -floor(-n)

A: In the given question we have to prove that ceil(n)=-floor(-n) That is…

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

A:

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: determine the number of computer can randomly generate one such integers from 1 through or more 12.…

A:

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: Show that for any nez, n is erther odd or even

A: Suppose P(n) is a given statement, then P(n) is true for all positive integer if: P(1) is true. If…

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: 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: 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: How many subsets of set S = {1, 2, 3, 4, 6} do not contain odd numbers? Use bijection to solve this…

A: Solution S={1,2,3,4,6}

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: Consider 4 digit numbers with each digit from the set {1,2,3,4,5,6,7,8,9} a) How many such numbers…

A: Solution

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

A:

Q: If n = 3, what is 3n? Submit

A:

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: Prove the fallowing egudion for every n,2 by hathematical enduelion. 35.6+35 6+ 35.63+ -+35.6"= 7.6"

A:

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

A:

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: Use a direct proof to show that the sum of two odd integers(k and l) is even.

A: The number n is even if it is expressed as  n=2m , where m is an integer. The number n is odd if it…

Q: How many integers between 1 and 10000(inclusive) are divisible by at one of 9, 10, 11?

A:

Q: If n is an integer greater than 2, then for all integers m, n does not divide m or n + m # nm.

A: c Given n∈Zand n>2. We have to prove that for all m∈Z, n/m or n+m/nm. Suppose n/m and…

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: 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: 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: Suppose that 31% of McMaster undergraduate students are in the first year. President David Farrar…

A: It is given that p=0.31.

Q: : Use he comparism test to prove hat n-2 Converyes

A:

Q: Suppose that you toss a fair coin 10,000 times. Write down explicitly the at the coin lands on heads…

A: Given information: Given that a fair coin is tossed 10,000 times.

Q: Let x and y be positive integers. When is x + y3 a prime number? When is x+y a composite number?…

A:

Statements and Logic

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