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Practice Test 5

(1) [3 points] If and are integers then if and only if is
    
    (a) always true.
    
    (b)  true if and are positive.
    
    (c)  true if and are negative.
    
    (d) true only if

    (e)  always false.

(2) [3 points] Given the function with domain then which of the following is true:

    (a) is not prime

    (b) is prime                
    
    (c) is prime and is not prime

    (d) is not prime    and is not prime            
    
    (e) is prime and is not prime
    

(3) [3 points] What is the greatest common divisor of and
    
    (a) or
    
    (b)   or
    
    (c)  1 or
    
    (d) or 2

    (e)  only 1

(4) [3 points] Given two nonzero integers and then means

    (a) and         
    
    (b) and
    
    (c) , , and if is a common divisor of and then
    
    (d) , , and if is a common divisor of and then
    
    (e) , , and if is a common divisor of and then
    

(5) [3 points] Every integer can be written as the product of possibly a square and a square-free integer is a consequence of which theorem: (A square-free integer is an integer that is not divisible by any perfect square other than 1).
    
    (a) Euclidean Algorithm
    
    (b)  Infinitude of Primes
    
    (c)  Fermat's Theorem
    
    (d)  Fundamental Theorem of Arithmetic

    (e)  Euler's Theorem

(6) [3 points] To solve the linear Diophantine equation we
    
    (a) perform the Euclidean Algorithm to find and such that
    
    (b) determine there are no solutions because
    
    (c) determine there are 6 incongruent solutions since
    
    (d)  perform the Euclidean Algorithm to find and such that and then multiply by 25.

    (e) determine there are 50 incongruent solutions and we use the Chinese Remainder theorem to find them

(7) [3 points] For which positive integers is the congruence equation true?
    
    (a) 22
    
    (b)
    
    (c)
    
    (d) 5

    (e) 1 and  5

(8) [3 points] For which integers with is solvable?
    
    (a)
    
    (b)  
    
    (c)  
    
    (d)  

    (e)  

(9) [3 points] When finding an integer that leaves a remainder of 1 when divided by either 2 or 5, but that is divisible by 3 we use
    
    (a) the Euclidean algorithm to show no such integer exists
    
    (b) the Euclidean algorithm to show such an integer exists
    
    (c) the Chinese Remainder to show no such integer exists
    
    (d) the Chinese Remainder to show such an integer exists

    (e) the Fundamental Theorem Arithmetic to factor 21.

(10) [3 points] Suppose is a solution to the polynomial congruence In an attempt to lift to a solution for we
    
    (a) always guess and check for
    
    (b) find the inverse for working   
    
    (c) factor each of the coefficients of   
    
    (d) compute the gcd of the coefficients of

    (e) compute the derivative of

(11) [3 points] The highest power of that divides is
    
    (a) 13
    
    (b)  5
    
    (c)  7
    
    (d)  8

    (e)  11

(12) [3 points] On April 13, 2029 it is known that the asteroid 2004 MN4 will have a 2.2% chance of hitting planet Earth. Determine the day of the week.
    
    (a) Monday
    
    (b)  Tuesday
    
    (c)  Wednesday
    
    (d)  Thursday

    (e)  Friday

(13) [3 points] Find the least positive residue of modulo
    
    (a)
    
    (b)  
    
    (c)  4
    
    (d)  5

    (e)  6

(14) [3 points] Find the last digit in the decimal expansion of
    
    (a) 1
    
    (b)  3
    
    (c)  7
    
    (d)  8

    (e)  9

(15) [3 points] Which of the following statements is FALSE:
    
    (a) is even provided is a positive integer
    
    (b)   provided and are positive integers
    
    (c) provided is a prime and is a positive integer
    
    (d)   provided is a prime integer

    (e) provided and are positive integers

(16) [10 points] Prove whenever is a positive integer.

(17) [10 points] Solve the congruence by writing a linear Diophantine equation and solving it.

(18) [10 points] Solve the congruence by using Euler's theorem.

(19) [10 points] What is the remainder when is divided by

(20) [15 points] Show that

(Extra Credit) An open conjecture of Carmichael asserts that for every positive integer there is another  positive integer such that Gather as much evidence as possible for this conjecture.

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