Pigeonhole Principle Prove that among any 100 integers there always two whose difference is divisible by 99.

Q: 2) x" = (Power RUle, n any nonzero integer)

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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: How many positive integers less than 850 are multiples of 8, 3, or 5? Use the Principle of…

A: Using the Principle of inclusion/Exclusion, the formula for A∪B∪C can be written as:…

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

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Q: A number N is divisible by 4 iff the number formed by the last 2 digits of N is divisible by 4. For…

A: We can find number divisible by 4  by using the concept given in the question

Q: Discrete Math

A: According to the given information, Suppose that:

Q: All composite number can represent as product of primes numbers

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Q: Suppose 48 cars start at a car race. In how many ways can the top 3 cars finish the race? The number…

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Q: The integers 77 and 105 are relatively prime. صواب İhi

A: By finding the factors of given numbers you can find the solution :

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

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Q: Let Sbe the set of all (nonnegative) 4-digit integer. How many integers from start with 1 and end…

A: let S be the set of all 4-digit integer. how many integers from start 1 and end with 9

Q: Fermat's theorem is not true for some prime number. Select one: O True O False

A: Fermat's theorem states that if p is a odd prime number, then for any integer a, the number ap − a…

Q: Find 1000 consecuticve composite integers. (Do not write them out).

A: To find :  The  1000 consecutive composite integers.      Formula Used :   The n consecutive…

Q: Suppose 50 cars start at a car race. In how many ways can the top 3 cars finish the race? The number…

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

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Q: noosing a positive integer from 100 to 999, what is the poSSibility of choosing a positive integer…

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Q: How many positive integers between 50 and 100 a) are divisible by 7? Which integers are these? b)…

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Q: Prove a number is divisible by 3 if and only if the sum of its digits is divisible by 3.

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Q: Show that, selecting 91 positive integers guarantees that there are at least 10 integers with the…

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Q: Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder 1, 5 when…

A: To find- Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder…

Q: 5. Give a recursive definition of the set of positive integers that are multiples of 5.

A: Solution

Q: Computing the greatest common divisor of two nonnegative integers a and b with a < b using recursive…

A: This is a problem of Number Theory, Modular Arithmatic.

Q: A student in COBU260 course scores 60 points out of a possible 110 in the first half of the…

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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: Describe a general method for finding all the divisors of a positive integer once you know the prime…

A: Given  A positive Integer whose prime factorization is given . To find all the divisors with general…

Q: How many consecutive zeroes are there at the end of 100! (100 factorial). For example, 12! -…

A: How many consecutive zeroes are there at the end of 100! We know that…

Q: identify all natural numbers having exactly 28 divisors.

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Q: 12 belongs to the set of natural numbers

A: The natural numbers (i.e. counting numbers) are numbers which are used for counting and ordering.…

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: A student in COBU260 course scores 110 points out of a possible 110 in the first half of the…

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Q: what is the smallest positive integer greater than 1 that divides the number 7^2021-13^2021

A: here 7×2021-13×2021  =2021(7-13)                                       =2021(-6)…

Q: If a and b are integers with gcd(a, b) =d Then gea (-) -- gcd

A: False

Q: Prove the “rule of 9”: an integer is divisible by 9 if and only if the sum of its digits is…

A: See the attachment

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: Prove: Any positive integer can be represented as an aggregate of different powers of 3, the terms…

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Q: Give a recursive definition for the set of all the positive integers that are less than 100 and not…

A: Given: Set of all positive integers that are less than 100. To find: Recursive definition for the…

Q: sum of all positive integers multiples of 8 between 9 and 99

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Q: 4. A property that, two consecutive positive integers are relatively prime. Consider any two…

A: Two consecutive numbers are said to be relatively prime or coprime numbers if they have no common…

Q: Check all the numbers that are divisors

A: The divisors of 12,210 are 2,3 and 6 .

Q: Give 5 examples to: ➢ Integers ➢ Natural numbers ➢ Irrational Numbers ➢ Rational Numbers and show…

A: Given: ➢ Integers➢ Natural numbers➢ Irrational Numbers➢ Rational Numbers

Q: 3. There are seven students who always compare their CS test grades. They discovered that either the…

A: It is given that there are seven students who always compare their CS test grades. They discovered…

Q: How many positive integers less than 2ˆ{20} are not divisible by 7, 11 or 13. Use…

A: The solution is given below

Q: 4 There is an infinite number of ways to represent 1 on the number line. In the space below vrika au…

A: We can express 1 in an infinite number of ways Have to write the four expressions multiplying by 1…

Q: How many twin primes are there where both primes are less than or equal to 31? Number of twin…

A: Twin prime: If there is only one composite number between two prime numbers, then that two numbers…

Q: Prove using Direct Proof: For all integers a, b and c if a|b and b|c then a|c.

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Q: any set of 201 integers must contain at least 21 elements that all have the same last digit.

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Q: Fermat's theorem is not true for some prime number. اختر واحد

A: False

Q: 9. Prove that the greatest common divisor of two positive integers divides their least common…

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