Number Theory Midterm to Final
Study Guide
Review Session – Thursday May 6 in IT 222
Key definitions/theorems (and section numbers) to know:
2.4 LCM – Lowest Common Multiple and how it relates to the gcd and the prime
factorization of the numbers involved.
3.1 Definition of prime
Fundamental theorem of Arithmetic understand that it says that every number factors into
a product of primes and that in canonical form, this factoring is unique.
Know Theorem 3.1: If p is prime and p|ab then p|a or p|b
Know how to show the square root of a prime is irrational (3.1)
Definition of what it means for two sets to have the same cardinality (handout)
Know the cardinalities of sets like: evens, odds, integers, rationals, irrationals, reals, power
set of … (handouts)
Understand digitalization arguments (handouts)
Be able to do the sieve and describe what to do (3.2)
Euclid’s theorem (Theorem 3.4) (Infinite number of primes)
Define Twin primes (Section 3.3) What do we know about them and what do we not know
about them?
State the Goldbach conjecture. (Section 3.4) every even integer above 4 is the sum of 2 odd
primes.
Define congruence (Section 4.2)
a  b(mod n)
Be able to calculate in this number system.
Define complete set of residues (Section 4.2)

Know theorem 4.2 Know that the first 3 statements tell us congruence is an equivalence
relation.
Know all about equivalence relations (handout):
1.Know what a relation is (a rule for making ordered pairs in a set S).
2. Know examples of relations that pass some of the properties but fail others. So be able to
say the rule for a relation that has whatever properties are desired, like, a relation on the
integers might be: aRb if a-b<10. (So that one is reflexive, not symmetric and not
transitive.)
Be able to use theorem 4.2 to decide questions of what divides what. (Section 4.2)
Understand positional notational systems for integers (Section 4.3) So any question about
the digits of a number is about starting with the representation:
AnyInteger  an b n  an1b n1 ...a1b1  a0
On the final, you are guaranteed to have a problem using this and its most likely the base
will be 10.

So guaranteed on the final – using the above - Know theorem 4.4 and how this lets us
reduce problems by modular arithmetic as in theorems 4.5, 4.6 and HW #27 4.5 – 9’s divide
4.6 11’s divide.
Other Practice problems:
First are from the take-home quiz:
1. Prove that if
** Proof: If
a  bmod n then gcd(a,n)  gcd(b,n)
a  bmod n then for some k, a  b  kn or a  kn b
Euclidean Algorithm
The
is based on this lemma (section 2.4): If
a  qb r then
variable
names, this says:
gcd(a,n)  gcd(b,n). Using different


If a  kn b then gcd(a,n)  gcd(n,b) Done.



2. Find the d=gcd(1155,910) and write d as a linear combination of 1155 and 910.
**Using
the above lemma repeatedly:
1155=1*910+245
910=3*245+175
245=1*175+70
175=2*70+35
70=2*75+0
So d=35 Now work backwards to get the linear combination:
35=175-2*70
= 175-2(245-1*175)=-2*245+3*175
=-2*245+3*(910-3*245)
=3*910-11*245
=3*910-11*(1155-1*910)
=-11*1155+14*910
3. Do the steps below: (See the RSA handout for why anyone wants to do this.)
a. Pick any two different primes that are over 20 and under 80. (Call them p and q).
** You can use your own numbers I’ll use p=23 and q=29
b. Find N=pq and the call Phi=(p-1)(q-1) and figure out Phi
**N=867 Phi=22*28 = 616
c. Find a value E>10 such that gcd(E,Phi)=1
** Let E = 3
d. Solve for D such that ED=1 mod 616
** So Find D such that ED-1=K616 or 1=3D-K616 so 1 is gcd (3,616) and we want, as above
the linear combination. You work it out for yoiurs just like in 2.
4. A return to middle school: prove that equivalence of fractions is, actually, an equivalence
relation. That is, assume a fraction is defined to be an ordered pair of integers (a,b)
(written
a
a c

b ) where b is not 0. Define the relation b d
** Reflevive property: Let
 a a

b b

since ab=ab

a
b
if and only if ad=bc.
be a fraction with a non-zero denominator.Then

Symmetric property: Assume
the same cross product.
Transitive property: Assume

a c

b d
a c

b d
Then we know ad=bc. But then
and
c e

d f
c a

d b
since it has
with b,d, and f not zero. So

ad=bc and cf=de. So adcf=bcde. (multiplying lefts and rights times each other.) So
If c=0 then a=0 by the fundamental theorem or algebra. By the same theorem e=0 so in this
a e

b f
case,


since both cross products are 0.
If c is not zero, divide both sides of adcf=bcde by cd to get af=be so

a e

b f
5
5
5
5
5. Find the remainder when 1  2  3  ...100 is divided by 4.
** We note that we want to find the above sum mod 4. By theorem 4.2d, this is the sum of

each term mod 4.
Note that any even
 number raised to the 5th power is divisible by 4. This each of these
terms mod 4 is 0. So we need only consider the sum
15 mod 4  35 mod 4  55.mod 4.. 975 mod 4  995 mod 4
Each of these odd numbers is either one more than a multiple of 4 or 3 more than a
multiple of 4. If we consider 4k+1,

Note:
And


4k 1  1mod 4 so by Theorem 4.2f 4k 1 15 mod 4 1mod 4
5
4k  3
5
 35 mod 4  243mod 4  3mod 4
So the above sum is equal to: (again using Theorem 3.2d)

1 31 3...1 3mod 4  0

New ones:
6. Give an example of a relation that is symmetric but is neither reflexive nor transitive.
** There are many possibilities. Try (“not equal to”).
2 85 is divided by 17.
7. Find the remainder if
** Note
2 4  16  1mod17 So by Theorem 4.2f,
 
21
284  2 4  1

21
mod17  1mod17
So applying Theorem 4.2d,
285  2mod17  15mod17 So the remainder is 15.
8. Show that the sum of twin primes (both greater than 3) is divisible by 12.

** By our quiz theorem, we need only show that 4 divides the sum and 3 divides the sum.
Suppose our twin
 primes are p and p+2. We know they are both add, so p=2k+1 for some k
and we conclude the other p+2=2k+3So the sum is 4k+4, so 4|(p+(p+2).
For 3, we know neither is divisible by 3 so p cannot be 3j for any j, and, in addition p cannot
be 3j+1 for any j because if it were the other prime, p+2 would be 3j+3 which is a multiple
of 3. So p=3j+2 for some j and we see the other prime p+2=3j+4, so their sum is 6j+6 which
is divisible by 3. Done.
9. Prove that if p is a prime and a and n are integers and
n n
n
n  1 and that p | a n then
gcd(p ,a )  p
n
n
n
** Proof. It is clear that p | p and that the gcd cannot exceed p So we need only show


n
n
p | a Consider the canonical prime factorization of a  p1 p2 ... pm (repeats allowed).
n
n n
n
Then the canonical prime factorization of a  p1 p2 ...pm . So if p divides this, then for
n
n
n
some k, pk p So pk  p is one of those factors
of a Done.




the next integer after it are
 quite interesting in term of
10. For n>2, the number n! and
factors of n! are less than or equal to n while
prime factorization:
Show that the only prime

the only prime factors of n!+1 are greater than n.
** Part 1. Consider the prime factors of n! n(n 1)(n  2)...2 1 By Corollary 1 of p. 40,
if p divides n!, then p divides one of these factors so p is less than or equal to it and thus less
than or equal to n.

Part 2. Proof by contradiction. Suppose p is a prime with p  n and p | (n!1) Since n!
is the product of all natural numbers up to n, p must be a factor of n!. But then p divides
both n! and n!+1, So p | (n!1)  n!1So p=1. Contradiction.



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