Find the number of four non-negative integer ordered tuples (x1, x2, x3, x4) such that xi + x2 + ¤3 + x4 = 16 and x<6, x2 < 6, x3 < 6 and x4 < 6

Q: 3. How many 4-digit positive integers are there that are even or contain no O's?

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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: Prove that 6 divides n^3 − n whenever n is a non-negative integer

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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: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

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

A: The numbers divisible by 4 is:

Q: 1. Show that subtraction is not associative for the set of integers.

A: Definition :Let "." be a binary operation and A is a setThen "." is said to be associative for A if…

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&gt;m. Then, there…

Q: Find the number of ways a 5-digit positive integer can be made such that in each of them every digit…

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

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Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n&gt;1. All prime divisors of n!+1 must be bigger than n. This is…

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

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

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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: 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: Show that if seven integers are selected from {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, there must be…

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Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

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: 32. Prove that 3 divides n + 2n whenever n is a positive integer.

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Q: If the sum of any two integers is even, then so is their difference.

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: Let A be the set of all positive integers whose last two digits form a number that is evenly…

A: We have the given condition as   Let us consider A be the set of all positive integers whose last…

Q: Show that if any 14 numbers are selected from the set S = {1,2, 3, ..., 25}, %3D there are at least…

A: Given statement is If any 14 numbers are selected from the set S = {1,2,3,...,25}, there are at…

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: 2.4 a. Find all the cluster points of the set Q of all rational numbers, and justify your…

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Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: 2. Classify as integer or non-integer. • 28 • -58 • 7/16 • 7/21 • 21/7

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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: 2. Possible whenn A- Zero b. Negutive integer C- Negative Yational d- Positive integer

A: We know that Laplace transform of

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: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

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: Use a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

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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: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

Q: 3. The v77 is between what two integers?

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Q: 2. Find a counterexample to the statement n2 – 4 - V nonzero integers n, is not an integer.

A: 2. Given statement is, ∀ nonzero integers n, n2-4n is not an integer

Q: Prove that it is impossible to choose three distinct integers (a – b)|(b – c), (b – c)|(c – a), (c –…

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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: Find all positive integers n such that both n + 2020 and n + 2021 integers. are LVn +1]

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Q: If X is a positive integer divisible by 6, and X <60 What is the greatcst possibe vaive of X 7

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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…