91Ó°ÊÓ

Prove that for every positive integer \(n,\) there are \(n\) consecutive composite integers. [Hint: Consider the \(n\) consecutive integers starting with \((n+1) !+2 . ]\)

Prove or disprove that there are three consecutive odd positive integers that are primes, that is, odd primes of the form \(p, p+2,\) and \(p+4 .\)

Prove that a parity check bit can detect an error in a string if and only if the string contains an odd number of errors.

What are the quotient and remainder when a) 44 is divided by 8\(?\) b) 777 is divided by 21\(?\) c) \(-123\) is divided by 19\(?\) d) \(-1\) is divided by 23\(?\) e) \(-2002\) is divided by 87\(?\) f) 0 is divided by 17\(?\) g) \(1,234,567\) is divided by 1001\(?\) h) \(-100\) is divided by 101\(?\)

Which positive integers less than 12 are relatively prime to 12\(?\)

Encrypt the message GRIZZLY BEARS using blocks of five letters and the transposition cipher based on the permutation of \(\\{1,2,3,4,5\\}\) with \(\sigma(1)=3, \sigma(2)=5\) , \(\sigma(3)=1, \sigma(4)=2,\) and \(\sigma(5)=4 .\) For this exercise, use the letter \(X\) as many times as necessary to fill out the final block of fewer then five letters.

Which positive integers less than 30 are relatively prime to 30\(?\)

Decrypt the message EABW EFRO ATMR ASIN, which is the ciphertext produced by encrypting a plaintext message using the transposition cipher with blocks of four letters and the permutation \(\sigma\) of \(\\{1,2,3,4\\}\) defined by \(\sigma(1)=3, \sigma(2)=1, \sigma(3)=4,\) and \(\sigma(4)=2\)

Show that the binary expansion of a positive integer can be obtained from its octal expansion by translating each octal digit into a block of three binary digits.

The ISBN-10 of the sixth edition of Elementary Number Theory and Its Applications is \(0-321-500 \mathrm{Q} 1-8,\) where \(Q\) is a digit. Find the value of \(Q .\)