SECTION 5.1:  2,  12
SECTION 5.2:   1
SECTION 5.3:   5,  9a
SECTION 6.1:   2a, b, c

1.  Find the smallest positive integer n such that all of the digits of n (in base 10) are equal to 1
     AND  n is divisible by 49.

2.  Determine  ord₄₇(2),    ord₄₇(-2),   and ord₄₇(5).
      (You can save considerable time by noticing that 49 = 2 (mod 47) and that 125 = -16 (mod 47).  )

3.   Let N = 14¹⁹ + 1.   Show that  3 and 5 divide N.     Prove that if q is any other prime that divides N,
       then   q = 1 (mod 38).
 
 
 
 

Assignment 6 (due Monday, March 1st)

SECTION 4.1:   3,  4,  22,  23
SECTION 1.5:   30 by using congruences.
SECTION 5.1:   8,  9

1.  Suppose that m₁,  m₂,  m₃  are positive integers. Assume that m₁ and m₃ are both odd
     and that 3 does not divide m₂.   Suppose that n is an integer which satisfies the congruences

      n = 2 (mod m₁),    n = 3 (mod m₂),    n = 8 (mod m₃) .

      Let M =m₁m₂m_3.   Carefully prove that (n, M) = 1.   (Refer to the propositions that you use
       by their name or number.)

2.   Let N = 289¹⁵5²¹10¹⁹2⁴³. Find the remainder when N is divided by 99.
       (Suggestion:  99 = 9^.11. use congruences modulo 9 and modulo 11. Then use the
         Chinese Remainder Theorem.)

3.  By using congruences, determine the missing digit x in the following calculation:
         (48x2)(147) = 71x124.
 
 

Assignment 5 (due Monday, February 22nd)

1.  Is there a positive integer n such that n is divisible by 17 and such that the last six digits
     (in base 10) of n are 111111 ?     (Don't try to find such an integer n. You should justify
     your answer.)

2.  Is there a positive integer n such that n is divisible by 47 and all  of the digits of n  (in base 10)
     are 1's ?   (Justify your answer, but don't try to find such an integer n.)

3.  Let p be a prime.   Let a be an integer.  Prove that  a² =  1 (mod p) if and only if a = 1 (mod p)
     or  a = p-1 (mod p).

4.  Determine ord₁₃(a) for each integer a in the range  1 <  a  < 12.    Determine ord₁₅(2) and
      ord₁₅(11).     Determine ord₈(3) and ord₈(37).

From text:  3.3:  4a, b, c,     9,   10,   18 .
 
 
 
 

Assignment 4 (due Friday, February 12th)

1.  Find all solutions to the equation   330x + 192y = 30,  where x and y are integers.
     If  x and y satisfy this equation, what is the remainder that x gives when divided
     by 16?  What can you say about the possible remainders that y can give when
     divided by 16?

2.  Carefully prove that  ³\/35  is an irrational number.

3.  By using properties of congruences, find the remainder that 2¹¹3¹³7¹²79⁸²⁵ + (-3)¹⁷25² + 15!
     gives when divided by 13.

4.  Find the integers  a and b determined by  (6+\/35)⁸  =  a + b\/35.   The rational number a/b
     will be a very good approximation to \/35 .

From text:   3.1 :   2,   6,   10,   26,           3.2 :  2a, c, d, e,   8,   10   .
 
 

Assignment 3 (due Friday, January 29th)

1. Let p be a prime. Let a be an integer. Suppose that p divides a³ . By using Euclid's
     Lemma,  prove that p³ divides a³ .

2.  Let a, b, and c be nonzero integers. Assume that (a,b)=1 and (b,c)=1.   Can you
     conclude that (a,c)=1 ?  Can you conclude that (ac,b)=1 ? Justify your answers.

3. The Diophantine Equation  x² - 35y² = -19 has solutions where x and y are positive
     integers.  One solution is x=4, y=1. By experimentation, find another solution where x
     and y are positive integers.  (It turns out that this equation actually has infinitely many
     such solutions.)    Suppose that x=a and y=b is a solution to the above equation (where
     a and b are integers).   Prove that (a,b)=1.

4.  Notice that 5,  11,  17,  23 , and 29 are primes. Are there any primes p > 5 with the
     property that  p+6, p+12, p+18, and p+24 are also primes? Justify your answer
     carefully.
 

From Text:   2.2 -  2,   4a,c,         2.3 -  2,   22b,c,d,     24b,c,    38  .
 
 

Assignment 2  (due Friday, January 22nd)

1.  Show that the following statement is false:       If n is a prime,  then  2ⁿ - 1 is prime.

2.  Prove that the equation  x² - 35y² = 19  has no solutions where x and y are integers.
     (Hint:  In class, we proved that the equation  x² - 35y² = 17 has no solutions where
       x and y are integers. Try to imitate that proof.)

3.  It is conjectured that infinitely many primes p exist such that p+4 is also prime.  It is
     also conjectured that infinitely many primes p exist such that p+8 is prime.
     Find all primes p such that both p+4 and p+8 are prime. Justify your answer.

SECTION  1.9:  12a, b,   14a,b,c
SECTION  2.1:  2,   4,  5,  8,  22
 
 
 
 
 
 
 

Assignment 1 (due Monday, January 11th)

    SECTION 1.3:   8,  12,  16,  44
    SECTION 1.5:   5,  6,  10,  12,  24,  30,  32
    A:  Factor 301 as a product of primes.  Find the closest integer to 301 which is a prime.
    B:   Suppose that   n, a  and b  are positive integers.  Consider the following statement:

                        Statement:  If n divides ab, then n divides a  or  n divides b.

           Give the contrapositive and the converse of the above statement.
           Is the above statement true?  Is its contrapositive true?  Is its converse true?
           In each case, give a proof or a counterexample.