Math 430
Midterm I Information
Spring 2015
WHEN: Monday, March 9, in class (no notes or books)
COVERAGE: The midterm will cover the material discussed in lecture from
Sections 2.1-2.5,3.1-3.3,4.2-4.4,5.2-5.3.
Note, you are only responsible for material that we explicitly discussed in lecture from the sections
above. Consult your class notes on this.
REVIEW SESSIONS:
Saturday, March 7: 4–6 PM (MH-615)
Sunday, March 8: 4–6 PM (MH-615)
STUDYING: Here is an overview of the topics we have covered. You should be comfortable with
all of the following words below:
Chapter 2: Divisor, divisible, Division algorithm, greatest common divisor, relatively prime, Euclid’s lemma, Euclidean algorithm, least common multiple, Diophantine equations
Chapter 3: Primes, Fundamental Theorem of Arithmetic, unique factorization, infinitude of primes
of various forms
Chapter 4: Congruence modulo n (and properties), special divisibility tests, linear congruences,
Chinese Remainder Theorem
Chapter 5: Fermat’s Little Theorem, absolute pseudoprime (Carmichael numbers), Wilson’s Theorem, quadratic congruences
THEOREMS FOR THE TEST: I would like you to carefully learn and understand the proofs
of some of the major results in number theory in this course. Below are listed 5 major theorems
in this course that we have discussed. On the first midterm, four of these theorems will appear (I
will state them on the exam for you exactly as they are stated in the text), and you will be asked
to supply a proof of any choice of two of them. Any correct proof is acceptable (i.e. proof given in
class, proof in book, or any other correct proof you would like to use). The proof given in class is
recommended.
• Theorem 2.3 (Greatest Common Divisor as a Linear Combination)
• Theorem 3.2 (Fundamental Theorem of Arithmetic)
• Theorem 3.6 (Infinitely Many Primes of the form 4n + 3)
• Theorem 4.8 (Chinese Remainder Theorem)
• Theorem 5.1 (Fermat’s Little Theorem)
IMPORTANT THEOREMS: In addition to the above theorems, here are some additional theorems that I will not ask you to prove, but you nonetheless should know them, and how to apply
them:
• Basic properties of divisibility (Theorem 2.2, especially part (g))
• Linear combinations of a and b are multiples of gcd(a, b) (Corollary , middle of page 22)
• Relatively primes integers (Theorem 2.4)
• Euclid’s Lemma (Theorem 2.5)
• Euclidean Algorithm (pages 26-27, including Lemma)
• Relationship between gcd and lcm (Theorem 2.8)
• Solutions to linear Diophantine equation (Theorem 2.9)
• Divisibility by a prime (Theorems 3.1)
• The infinitude of primes (Theorem 3.4–Euclid)
• Characterization of two congruent integers (Theorem 4.1)
• Properties of congruences (Theorem 4.2)
• Cancelling in a congruence (Theorem 4.3 and its corollaries)
• Divisibility tests for 9 and 11 (Theorems 4.5 and 4.6)
• Solutions to linear congruence (Theorem 4.7)
• System of congruences with mods not relatively prime (Variation #2 of the Chinese Remainder
Theorem, discussed in class)
• Corollary to Fermat’s Little Theorem
• Wilson’s Theorem (Theorem 5.4)
• Solution to quadratic congruence (Theorem 5.5)
THINGS TO BE ABLE TO DO:
• Prove divisibility statements (see problems in section 2.2).
• Prove statements about gcd and lcm.
• Apply the Euclidean Algorithm forward and backward to determine gcd.
• Solve Diophantine equations.
• Solve word problems that can be formulated as Diophantine equations.
• Find prime factorizations.
• Use the uniqueness of prime factorizations to prove things (see Theorem 3.3, problems on
Group Work 3, and the problems in section 3.1).
• Determine whether certain integers are prime or not.
• Show that there are an infinitude of primes of a certain form (see Euclid’s proof, Theorem 3.6,
and problem 2 of Group Work 2, for example).
• Prove basic properties of congruences.
• Solve linear congruences, if possible.
• Create and carry out divisibility tests for particular integers.
• Use the C.R.T. to solve systems of congruences. If necessary, apply Problem 11 in Section 4.4
to handle systems with mods that are not relatively prime.
• Apply Fermat’s Little Theorem and Wilson’s Theorem for specific primes. This may involve
decomposing a composite integer into a prime factorization (see the examples with absolute
pseudoprimes).
ADVICE: I suggest reviewing the group work and the quiz very carefully. If you have time, it might
be best to try re-working those exercises from scratch and looking up the answers afterwards. You
will not have time to re-do all of the homework, but you might try some of them again, especially
the ones that will assist you with the “THINGS TO BE ABLE TO DO” listed above. Try to re-do
problems from scratch, rather than just looking up solutions that I’ve created. Focus more on the
mainstream stuff and the frequently applied techniques and methods for solving, NOT weird special
cases or tricks that don’t come up. I will not write the test with the intention of making it tricky!
If you prefer, you can just try some extra problems from scratch. I’ve included a number of random
practice problems below that you can try to work. I will post solutions to them as the midterm
approaches.
It may also help you to discuss solutions to problems with other people. Explaining yourself to
someone else can be a tremendous benefit to both of you, especially with this particular material.
Often, there is more than one way to look at things, so it is useful to hear each other’s perspective
on the problems at hand.
Finally, you can ask me questions as much as you want, and I will be happy to review or pop-quiz
a topic with you if you feel shaky. Basically, I’m here to help and I want everybody to do well, so
please don’t be shy :=) !!
SAMPLE PROBLEMS:
Problem 1: Find the last digit of the number
1919
1719
.
Problem 2: Determine all solutions in positive integers to the equations below:
(a): 54x + 21y = 906.
(b): 123x + 360y = 99.
Problem 3: If x ≡ a (mod n), prove that either x ≡ a (mod 2n) or x ≡ a + n (mod 2n).
Problem 4: A certain integer between 1 and 1200 leaves the remainders 1, 2, 6 when divided by
9, 11, 13, respectively. What is the integer?
Problem 5: Prove that if p ≥ q ≥ 5 and p and q are both primes, prove that
24|p2 − q 2 .
Problem 6. Prove that
√
5
11 is irrational.
Problem 7. For n > 3, prove that the integers n, n + 2, and n + 4 cannot all be prime.
Problem 8. Prove that the greatest common divisor of two integers divides their least common
multiple.
Problem 9. Solve the Diophantine equation
119x + 272y = 51.
Problem 10: Find the solutions to the system
11x + 5y ≡ 7
(mod20)
6x + 3y ≡ 8
(mod20)
Problem 11: Prove that if p and q are distinct primes, then
pq−1 + q p−1 ≡ 1
(modpq).
Problem 12: Prove that for every integer n ≥ 0, 13|1112n+6 + 1 and that 43|6n+2 + 72n+1 .