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: 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: 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 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…