Problem 1E: For n=5 , 8, 12, 20, and 25, find all positive integers less than n and relatively prime to n. Problem 2E: Determine a. gcd(2,10) lcm(2,10) b. gcd(20,8) lcm(20,8) d. gcd(21,50) lcm(21,50) e. gcd (p2q2,pq3)... Problem 3E: Determine 51 mod 13, 342 mod 85, 62 mod 15, 10 mod 15, (8273) mod 7, (51+68) mod 7, (3524) mod 11,... Problem 4E: Find integers s and t such that 1=7s+11t ? t. Show that s and t are not unique. Problem 5E: Show that if a and b are positive integers, then ab=lcm(a,b)gcd(a,b) . Problem 6E: Suppose a and b are integers that divide the integer c. If a and b are relatively prime, show that... Problem 7E: If a and b are integers and n is a positive integer, prove that a mod n=b mod n if and only if n... Problem 8E: Let d=gcd(a,b) . If a=da and b=db , show that gcd(a,b)=1 . Problem 9E: Let n be a fixed positive integer greater than 1. If a mod n=a and b mod n=b , prove that... Problem 10E: Let a and b be positive integers and let d=gcd(a,b) and m=lcm(a,b) . If t divides both a and b,... Problem 11E: Let n and a be positive integers and let d=gcd(a,n) . Show that the equation ax mod n=1 has a... Problem 12E: Show that 5n+3and7n+4 are relatively prime for all n. Problem 13E: Suppose that m and n are relatively prime and r is any integer. Showthat there are integers x and y... Problem 14E: Let p, q, and r be primes other than 3. Show that 3 divides p2+q2+r2 . Problem 15E: Prove that every prime greater than 3 can be written in the form 6n+1or6n+5 . Problem 16E: Determine 71000 mod 6 and 61001 mod 7. Problem 17E: Let a, b, s, and t be integers. If a mod st=bmodst, show that amod s=bmods and a mod t=bmodt. What... Problem 18E: Determine 8402 mod 5. Problem 19E: Show that gcd(a,bc)=1 if and only if gcd(a,b)=1 and gcd(a,c)=1 . (This exercise is referred to in... Problem 20E: Let p1,p2,...,pn be primes. Show that p1p2pn+1 is divisible by none of these primes. Problem 21E: Prove that there are infinitely many primes. (Hint: Use Exercise 20.) Problem 22E Problem 23E Problem 24E: For any complex numbers z1andz2 prove that z1z2=z1z2 . Problem 25E: Give an “if and only if” statement that describes when the logic gatex NAND y modeled by 1+xy is 1.... Problem 26E: For inputs of 0 and 1 and mod 2 arithmetic describe the output ofthe formula z+xy+xz in the form “If... Problem 27E: For every positive integer n, prove that a set with exactly n elements has exactly 2n subsets... Problem 28E: Prove that 2n32n1 is always divisible by 17. Problem 29E: Prove that there is some positive integer n such that n, n+1,n+2, , n+200 are all composite. Problem 30E: (Generalized Euclid’s Lemma) If p is a prime and p divides a1a2an , prove that p divides ai for some... Problem 31E Problem 32E: What is the largest bet that cannot be made with chips worth $7.00 and $9.00? Verify that your... Problem 33E: Prove that the First Principle of Mathematical Induction is a consequence of the Well Ordering... Problem 34E: The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . . In general,the Fibonacci numbers are... Problem 35E: Prove by induction on n that for all positive integers n, n3+(n+1)3+(n+2)3 is a multiple of 9. Problem 36E: Suppose that there is a statement involving a positive integer parameter n and you have an argument... Problem 37E: In the cut “As” from Songs in the Key of Life, Stevie Wonder mentions the equation 888=4 . Find all... Problem 38E: Prove that for every integer n, n3 mod 6=n mod 6. Problem 39E: If it is 2:00 A.M. now, what time will it be 3736 hours from now? Problem 40E: Determine the check digit for a money order with identification number 7234541780. Problem 41E: Suppose that in one of the noncheck positions of a money order number, the digit 0 is substituted... Problem 42E: Suppose that a money order identification number and check digit of 21720421168 is erroneously... Problem 43E: A transposition error involving distinct adjacent digits is one of theform ...ab......ba... with ab... Problem 44E: Determine the check digit for the Avis rental car with identification number 540047. (See Example... Problem 45E: Show that a substitution of a digit ai for the digit ai(aiai) in a noncheck position of a UPS number... Problem 46E: Determine which transposition errors involving adjacent digits are detected by the UPS check digit. Problem 47E: Use the UPC scheme to determine the check digit for the number 07312400508. Problem 48E: Explain why the check digit for a money order for the number N is the repeated decimal digit in the... Problem 49E: The 10-digit International Standard Book Number (ISBN-10) a1a2a3a4a5a6a7a8a9a10 has the property... Problem 50E: Suppose that an ISBN-10 has a smudged entry where the question mark appears in the number 0716?28419... Problem 51E: Suppose three consecutive digits abc of an ISBN-10 are scrambled as bca. Which such errors will go... Problem 52E Problem 53E: Suppose the weighting vector for ISBN-10s were changed to (1, 2,3, 4, 5, 6, 7, 8, 9, 10). Explain... Problem 54E: Use the two-check-digit error-correction method described in thischapter to append two check digits... Problem 55E: Suppose that an eight-digit number has two check digits appended using the error-correction method... Problem 56E: The state of Utah appends a ninth digit a9 to an eight-digit driver’slicense number a1a2...a8 so... Problem 57E: Complete the proof of Theorem 0.8. Problem 58E: Let S be the set of real numbers. If a,bS , define a~b if ab isan integer. Show that ~is an... Problem 59E: Let S be the set of integers. If a,bS , define aRb if ab0 . Is R an equivalence relation on S? Problem 60E: Let S be the set of integers. If a,bS , define aRb if a+b is even. Prove that R is an equivalence... Problem 61E: Complete the proof of Theorem 0.7 by showing that ~is an equivalence relation on S. Problem 62E: Prove that 3, 5, and 7 are the only three consecutive odd integers that are prime. Problem 63E: What is the last digit of 3100 ? What is the last digit of 2100 ? Problem 64E: Prove that there are no rational numbers x and y such that x2y21002 . Problem 65E: (Cancellation Property) Suppose , and are functions. If =and is one-to-one and onto, prove that = . format_list_bulleted
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