Number Theory Practice Test 1

Problem (1) Using Mathematical Induction to prove, a correct step would be:

    (a) If for all then ...   

    (b) If for some then ...     

    (c)     

    (d) If for all   then ...    

    (e) none of the above

Problem (2) Using Mathematical Induction to prove, is divisible by a correct step would be:
    (a) If for all then ...       
    (b) If for some then ...
    (c) If then ...       
    (d) If for some then ...        
    (e) none of the above

Problem (3) The product of two integers of the form is of the form because:
    (a)   for all
    (b)   for some    
    (c)  actually, the result is not true.    
    (d)   for some    
    (e)  none of the above

Problem (4) The product of three consecutive integers is divisible by 6 because
    (a) actually, the result is not true.   
    (b) because one of and might be divisible by 6.
    (c) because one of and is divisible by and one is divisible by
    (c) because and are all divisible by and    
    (e) none of the above

Problem (5) Any prime of the form is also of the form because
    (a) using the division algorithm we can write any prime in the form or and so the result is not true.    
    (b) using the division algorithm we can write any integer in the form or and so the result is not true.  
    (c) using the division algorithm we can write any integer in the form or and then we can show that is not prime.    
    (d)  using the division algorithm we can write any integer in the form or and then we can show that is not prime.  
    (e) none of the above

Problem (6) There are infinitely many primes
    (a) was proven by Euclid     
     (b) actually, the result is not true.  
     (c) because you can always add a one and get another prime   
     (d) by the Sieve of Eratosthenes   
     (e) none of the above

Problem (7) True or False: divides for any integer and any positive integer
    (a) True   
     (b) False    
     (c) none of the above

Problem (8) For any integer , because
    (a)    
    (b)     
    (c)  not true for
    (d) and so actually   
    (e) none of the above

Problem (9) Select the one(s) that are false,  
    (a)    
    (b)     
    (c)     
    (d)      
    (e) none of the above

Problem (10) Apply the Euclidean Algorithm to find
    (a)      
    (b)       
    (c)      
    (d)         
    (e) none of the above

Problem (11) The power of in the unique factorization of   is
    (a) 2      
    (b) 3      
    (c) 1     
    (d) not a factor      
    (e) none of the above

Problem (12) The primes in the unique factorization of   are
    (a)       
    (b)       
    (c)      
    (d)       
    (e) none of the above

Problem (13) Determine how many solutions to
    (a) No solution because        
    (b) No solution because
    (c) Infinitely many solutions because
    (d) Infinitely many solutions because      
    (e) none of the above

Problem (14) Find all solutions to
    (a) No solution because        
    (b) Only one solution of and
    (c) Infinitely many solutions:   and
    (d)  Infinitely many solutions: and       
    (e) none of the above

Problem (15) Show that if then for all positive integers .

Problem (16) Apply the Euclidean Algorithm to solve Then find all solutions.

Problem (17) Show that if and are mutually prime integers, then

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