show that for every positive integer n there is a prime whose decimal expansion contains n consecutive 1s and and whose final digit is 3

Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: 3n+2 and 5n+3 are relatively prime for every integer n.

Q: Compute for an arbitrary integer k

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Q: Use mathematical induction to prove that n3 - n is divisible by 3 where n is a positive integer.

A: We need to prove that  is divisible by  for all positive integer  using mathematical induction.We…

Q: at any positive odd integer is of the

A: Here we have to show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5,…

Q: Find the product of all positive integers from n down through 1 for n = 5.

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

A:

Q: If you randomly pick integers from 1 to 200, how many would you have to pick to ensure that at least…

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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 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: 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: Prove that the number in between twin primes is always a multiple of 6.

A: A Twin prime are those numbers which are prime and having a difference of two ( 2 ) between the two…

Q: Find the sum of the first five natural even numbers using sigma notation.

A: We have to find sun of first 5 even natural numbers.

Q: True or False Any whole number is divisible by 3 if and only if the sum of its digits is divisible…

A: We have to find the our statements are true or false

Q: Use generating functions to determine the number of 9-digit sequences of the digits 1,2,3,4,5,and 6…

A: We need to find ,  number of 9-digit sequences of the digits 1,2,3,4,5,and 6 in which each even…