Prove that the sum of two consecutive odd integers is divisible by 4.

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

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Q: plz solve q4

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Q: Prove that the product of any k consecutive positive integers is an integer multiple of k!.

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Q: Show that for all positive integers n, 13 + 2° + ...+ n³ = (1+2+3+ ...+ n)?

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Q: Prove the statement. "For any integer n, nº +4 is not divisible by 8." by Contradiction.

A: Given: n be any integer. To prove: n3+4 is not divisible by 8.

Q: Show that for any integer n, n² – 2 is not divisible by 3.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 4) Prove that the product of any three consecutive integers is divisible by 6.

A: We have to show that product of any three consecutive integers is divisible by 6

Q: Prove that 5 does not divide 5n+3 for any integer n.

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Q: Show that an integer N is divisible by 9 if and only if the sum of its digits is divisible by 9

A: Let a=anan-1...a3a2a1 be an integer and S be the sum of the digits. Consider a,…

Q: If p + 2 is a prime number, prove that p² + 3 is divisible by 4.

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Q: Prove that the sum of any three consecutive numbers is divisible by 3

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Q: Show that for any integer n , n2 – 2 is not divisible by 3.

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Q: Prove by using direct proof: Prove that the sum of any three consecutive integers is divisible…

A: Solution   Prove that the sum of any three consecutive integers is divisible by 3.

Q: Prove by contradiction : For any odd integer n : n² – 7 is not divisible by 4.

A: Proving a statement by contradiction means assume the negation of statement true and prove the…

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: Prove that for any odd positive integer, n, the sum of nconsecutive positive integers is divisible…

A: To prove for any odd positive integer, n, the sum of nconsecutive positive integers is divisible by…

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

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Q: Prove that 2^10 + 5^12 is a composite number.

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Q: Find the sum of the positive even integers up to and including 50.

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Q: Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

A: we have to Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the…

Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

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Q: Show that if n is a positive integer then =n(n + 1)2"-2

A: We know the binomial theorem formula isx+yn=∑k=0nCkn xn-kyk, where Ckn is the binomial co-efficient…

Q: Discrete

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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: Find the sum of all integers that are multiples of 3 from 4 to 100.

A: We have to find the sum of all integers that are multiples of 3 from 4 to 100 Then, If A denotes the…

Q: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

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Q: (a) Suppose that there are k distinct positive integers that divide an odd positive integer n.…

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Q: Prove that if the sum of the digits is subtracted from a natural number, then the result is…

A: let us assume the number to be three digit number xyz therefore, xyz=100x+10y+z sum of digits will…

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: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

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 that 4 does not divide n2 +2 for any integer n.

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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: Show by contradiction that for any integer n: if n>0, then n+10 n+5

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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 by contradiction that 2020 cannot be written as a sum of 3 odd integers.

A: The objective is to prove  that 2020 cannot be written as a sum of 3 odd integers by the…

Q: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

Q: Prove that 3" + 1 is divisible by 4 whenever n is an odd positive integer.

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Q: find the sum of all integers between 300 and 800 that are exactly divisible by 14

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Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

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Q: Show that every odd integer is the difference between two consecutive perfect squares.

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Q: Show that if n is an integer, thenn= [n/2] + [n/2].

A: Floor function x: x gives as output the greatest integer less than or equal to x. Ceiling function…

Q: Is it true that for any odd n the number 2" +1 is divisible by 3? Prove your answer.

A: We prove this by using PMI(Principle of Mathematical induction)

Q: Prove that 30 divides n° – n for every nonnegative integer n. -

A: The given question can be solved as shown in step2.

Q: Find the sum of all natural numbers n such that 2021-n/99 is a natural number ?

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Q: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

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Q: Prove by induction that the product of the first n even numbers is equal to n!2" for all positive…

A: We have to prove by induction that the product of first n even numbersis equal to n!2n for all…

Q: divisible

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!