Show that, selecting 91 positive integers guarantees that there are at least 10 integers with the same ones digit.

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: How many integers between 100 and 400 contain the digot 2?

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Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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

A: The numbers divisible by 4 is:

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: Find the number of positive integers X smaller than 12849 which have exactly 2022 positive divisors

A: Theory : Total no of divisor of a number, If its prime factorization, given by ∏i=1hpiai Where p is…

Q: Determine the number of natural numbers less than 700 that can be written using the digtits 5,6,7,9?

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Q: The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is

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Q: Show that a number with 30 digits cannot have more than 100 prime factors.  

A: Prime factor: A factor of a number which is a prime number. For example: consider a number 26. The…

Q: How many four digit positive integers are there that contain the digit 3 and are divisible by 5?

A: A number is divisible by 5, if the unit digit place contains the number 0 or 5. Let m choices to be…

Q: Any integer greater than 1 that has other positive factors besides 1 and itself is also known as O…

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Q: ive distinct integers are to be chosen from the set {1,2,3,4,5,6,7,8} and placed in some order in…

A: Given: To find the number of possible ways in which integers can be chosen,

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

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Q: of the first 150 positive integers?

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

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Q: Show that if any eight positive integers are chosen, two of them will have the same remainder when…

A: Pigeonhole Principle- If n items are assigned to m containers, and m<n,then at least one…

Q: Find how many positive integers with exactly four decimal digits, that is, positive integers between…

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Q: a) How many 3-digit positive integers are there with odd digits?

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Q: Which positive integers less than 20 are relatively prime to 20?

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Q: How many 5–digit (decimal) integers are there with none of the digits repeated?

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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: Find all three-digit numbers with different digits with the property that those numbers are…

A: Let N=100a+10b+c represent all the three-digit numbers, for some digits, a, b and c. Then,…

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: Discrete

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Q: How many integers greater than 120 but less than 540 are divisible by 5?

A: Given Problem:- How many integers greater than 120 but less than 540 are divisible by 5?

Q: How many integers between 1000 and 9999 inclusive are not divisible by 5?

A: The number of integers from 1 and 9999 which are divisible by 5 is = [9999 ÷ 5] = [1,999.8] = 1999…

Q: Compute P5,2. (Enter an exact number.)

A: Obtain the value of P5,2.  The value of P5,2 is obtained below:   Permutation: A permutation is an…

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: Suppose 59 cars start at a car race. In how many ways can the top 3 cars finish the race? ... The…

A: There is total 59 cars.

Q: Compute P5,4. (Enter an exact number.)

A: Obtain the value of P5,4.  The value of P5,4 is obtained below:   Permutation: A permutation is an…

Q: 21 and 24 are not relatively prime numbers

A: To find: Whether 21 and 24 are relatively prime numbers or not?

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

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Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

Q: Five distinct integers are to be chosen from the set {1, 2, 3, 4, 5, 6, 7, 8} and placed in some…

A: Given Information -  Five distinct integers are to be chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}…

Q: 5. Find the number of positive integers less than 700 that are divisible by 3 but not divisible by…

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

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Q: Suppose 26 cars start at a car race. In how many ways can the top 3 cars finish the race? ..... The…

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Q: Find all three-digit numbers with different digits such that those three-digit numbers are divisible…

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Q: Determine all natural numbers divisible by 8 whose sum of digits is less than 10 and the product of…

A: Permutation and combination concepts will be used to filter out all the 3 scenarios in getting…

Q: Show that the set of odd numbers is countable.

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Q: How many 5 digit positive integers are divisible by 5 and have at least one 6 as their digit?

A: A number is divisible by 5, if it has a 0 or 5 in the unit digit place. Let m choices to be made…

Q: How many positive integers not exceeding 10,000 are divisible by 7, 17, or 23?

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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: How many four-digit positive integers both start and end in even digits?

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Q: How many positive integers between 10 and 99 inclusive are divisible by 2 or 3?

A: How many positive integers between 10 and 99 inclusive are divisible by 2 or 3?

Q: If we try to get a 6-digit integer with non-repeated digits from 0,2,4,7,8,9, how many integers are…

A: We can use the permutation and combination in order to solve the given problem.

Q: How many corresponding integers are made up of 1443 digits taken from (1-2-3-4-5-6) provided that no…

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Q: How many four-digit positive integers both start and end in even digits:

A: We need to find the number of positive integers both start and end in even digits.