9. This question is a little more challenging. How many integers in the interval [1, 99 999] (i.e., the integers between 1 and 99 999 inclusive) have digits whose sum is 10. For example, 46

Q: Prove that Va - Vb is irrational number if 'a' and 'b’ are prime number?

A: Given To prove that a-b is irrational number if a and b are prime number.

Q: What is the set M of positive integers less than 100?

A: Given statement is set consist all those positive integers less then 100. Collection of all these…

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: 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: If ten integers are chosen from between 0 and 21 exclusive, must at least one of them be even?…

A: Pigeon hole principle states that if n objects are put in to m containers with n>m then at least…

Q: 2. Suppose that 21 girls and 21 boys enter a mathematics competition. Furthermore, suppose that each…

A: Consider a mathematics competition of 21 girls and 21 boys along with the condition that each…

Q: 17. The number of 3-digit numbers less than 500 that can be created if the last digit is either 4 or…

A: We have to form 3 digit numbers less than 500 with last digit is either 4 or 5

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: 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: Suppose that 43% of people own dogs. If you pick two people at random, what is the both own a dog?…

A: Given, Probabily of a person own a dog is 43 %

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: A company wants to have 6-digit phone extensions with the first digit not being zero. How many…

A: Given, No. of digit in phone extension = 6 First no- cannot be  0 Therefore no. of ways it can be…

Q: . This question is a little more challenging. How many integers in the interval [1, 99 999] (ie.,…

A: An equivalent formulation for the given problem: The number of integer solutions to the equation:…

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: The set of elements common to both (-*, 9) and (-*, 12) is of these intervals. . This represents the

A: The set of elements can be expressed in the interval form using open interval, closed interval,…

Q: 1. The set of counting numbers greater than -1 and less than 9 2. The set of integers satisfying X -…

A: Roster method is to express the set in terms of flower brackets, for example to write the set of…

Q: Prove the statement by the method of exhaustion: For each integer n with 1 less than or equal to n…

A:

Q: 1.How many license plates consisting of 6 digits (0 to 9) have their digits in strictly increasing…

A: Given that, How many license plates consisting of 6 digits (0 to 9) have their digits in strictly…

Q: How many pairs of two distinct integers chosen from the set {1, 2, 3,..., 101} have a sum that is…

A: Since you have posted multiple questions and we can answer one question. So answer provided for the…

Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

A:

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

A:

Q: This question is a little more challenging. How many integers in the interval [1, 99 999] (ie., the…

A: This is a star and bars problem Let a, b,c,d,e are positive integers where a +b+c+d+c =10 So, we…

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

A:

Q: Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it…

A: (a)The fill up for the statement to rewrite the given statement: Is there an integer n such that n…

Q: 3. If а, 1000". Find N E I, the set of positive integers, such that an+1 N). n!

A: Given an=1000nn!. We have to find the N∈I such that an+1<an, n>N. For n=1 a1=100011! For n=2…

Q: When the positive integer n is divided by 3, the remainder is 2 and when n is divided by 5, the…

A:

Q: : a number is even  if

A:

Q: What is the range of all integers greater than 40 and less than 99 that are multiples of 3

A: To find : The range of all integers greater than 40 and less than 99 that are multiples of 3

Q: How many pairs of two distinct integers chosen from the set {1, 2, 3, 101} have a sum that is even?

A:

Q: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

A: The given term is n2+5n

Q: There are 10 questions in a final ėxamination for Discrete Mathematics. Each question is assigned at…

A:

Q: 4. A witness to a hit and run accident told the police that the license number contained  the…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: 9. Show that 1729 is a Carmichael number.

A: To show that 1729 is a carmichael number

Q: How many integers between 100 and 400 contain the digit 2? There are integers between 100 and 400…

A: Numbers between 200 and 299 contains 2 So total integers = 100 Numbers between 120 – 129 contains 2…

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: Prove, that (nezond n73) If x"=5,+hen x is not a rational number.

A:

Q: 1. Show that 1105 is a Carmichael number.

A:

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: Rprove that for ony natural number n, 2+4+ --

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: ind the number of ways to choose k children from a group of n children arranged in a circle so that…

A: Consider the given: Let n and k be positive integers with . Choose the number of ways in which k…

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

A:

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

A: We have to prove

Q: Since the next prime after 7 is​ 11, the smallest composite number whose prime factors are all…

A:

Q: how that all the irrational numbers in the interval [0, 1] is dense there.

A:

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

A:

Q: How many words of lengthn≥6 in the alphabet{1,2,3}contain at least two of each digit?(Prove a closed…

A: Here we have to find the total number of possible ways of length n≥6 in the alphabet {1,2,3} contain…

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…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.