Math 58 – Elementary Number Theory
October 5, 2006
What’s on the Exam? #1
First-page header (subject to revision)
Name_____________________________
Math 58 – Elementary Number Theory
October 5, 2006
Exam 1 – Due at start of class Tuesday, October 10
Take this exam at a time and place of your choosing. There is no time limit, but you are enjoined
from discussing math with anyone except the instructor between starting and ending the
exam, so you should plan on completing it in a single session. You may use a calculator
but not a computer or any references. You may use this paper, your own paper, or both.
It isn’t necessary to turn in scratch paper. There are 15 numbered questions on 3 pages.
Show your work. (It may sometimes be necessary for full credit, and it’s always necessary for
part credit.)
Return this exam in class or to my office by 11:20am on Tuesday, October 10.
(1) Be able to do these computations…
compute the gcd of two or more numbers, of up to, say, 10 digits.
compute the lcm of two or more numbers, provided their product is at most,
say, 10 digits. (One method: [ x, y, z ] = xyz – ( x, y, z ). )
compute the gcd and lcm of two or more numbers much more quickly if
given their prime factorizations.
given a, b, n, decide whether n can be written as n = xa + yb for
for (positive or negative) integers x and y, and if possible, find
x and y
n
evaluate binomial coefficients,  
k 
convert an integer from decimal representation to binary, or vice versa
find the least non-negative residue of a (mod m)
given a and m, decide whether a has a multiplicative inverse (mod m), and
and if so, find a-1 (mod m)
find all x that satisfy ax = c (mod m), even if (a,m) is not 1
1
calculate products and exponents (mod m) without fear, e.g. ab or a! (mod m),
even when a, b are fairly large (also know some shortcuts for
these; see below)
Chinese remainder theorem: If m1, …, mr are pairwise relatively prime
and a1, …, ar are given, find x such that x = ai (mod mi) for
each i, and know that x is unique mod m1m2…mr. (The case in
which the m’s are not pairwise relatively prime is not on the exam.)
Find the prime factorization of n, if n is small enough that trial division
works
Find the prime factorization of n, if n is given in a form that permits
trickery, such as the difference of two squares
Given a prime p and a primitive root g for the integers mod p, list all of
the k-th powers (mod p). (They are 1, gk, g2k, g3k, etc…; the
exponents might stop at gp-1 or keep going for more cycles
according to whether k is a factor of p-1.)
(2) Notations
(a, b) = gcd of a, b
(a, b, c, …) = gcd of a, b, c, … = ( a, (b, c, … ) )
[a, b] = lcm of a, b
[a, b, c, …] = lcm of a, b, c, …
[a] = congruence class of a, if modulus is understood
all numbers are integers unless clearly stated otherwise, but we make
no automatic assumption about whether they are positive
primes are always positive
Zm = system of integers mod m. (Might be seen as not integers at all, but
system of congruence classes mod m with + and  defined for
congruence classes)
Zp* = system of non-zero integers mod p, if p is prime (Analogous
notation for non-primes not introduced yet)
(3) Some random theorems
Any common divisor of a, b, c,… is a divisor of (a, b, c, …)
Any common multiple of a, b, c… is a multiple of [a, b, c… ]
Division algorithm: Given integers a and b with b positive, there are unique
integers q, r such that a = qb + r and 0 ≤ r < b.
2
If a|bc and (a,b)=1 then a|c. If p is prime and p|bc, then p|b or p|c.
a has a multiplicative inverse mod m if, and only if, (a,m)=1.
(Wilson) (p-1)! = -1 (mod p) if, and only if, p is prime.
(Fermat’s Little Theorem) ap = a (mod p) if p is prime, for any a.
Equivalently: If p is prime and p does not divide a, then ap-1 = 1 (mod p).
Primitive roots: If p is a prime, there is an integer g in Zp such that all non-zero
elements of Zp are powers of g. (We have declared this to be true,
but we haven’t proven it yet.)
There are infinitely many primes
There are arbitrarily long sequences of evenly spaced primes (not proven
in class)
There are arbitrarily long sequences of positive composite numbers
The density of primes in the neighborhood of x is about 1/ln(x) (heuristic)
The number of primes less than x, called (x), is about x/ln(x),
x
1
dt . (True in limit; that is,
or more accurately Li(x) = 
t  2 ln t
lim (x)/Li(x) = 1, but not proved in class)
(4) Bloopers, rejects, and practice problems
1. State the “division algorithm.”
2. State precisely the fundamental theorem of arithmetic.
3. Evaluate: ( 8174, 12261) and [ 125, 15 ]
4. Express in base-10 notation: (1111000)2
5. Find the prime factorization of 6237.
6. What is the prime factorization of 4087 ? (Hint: 642 = 4096)
3
7. Express 7 as a linear combination of 15 and 4, or prove that it is impossible.
8. Express 7 as a linear combination of 15 and 50, or prove that it is impossible.
9. The 5th prime is 11. Prove by induction that if n ≥ 5, then the n-th prime is
larger than 2n.
10. Can there be 19 consecutive composite integers, all greater than 1 ?
(Prove your answer)
11. Find all integers x such that 10x = 9 (mod 64) or prove that no such
x exists.
12. Write a complete residue system (mod 8).
13. Find an integer x such that all of these congruences are true:
x=1 (mod 2) x=2 (mod 3) x=4 (mod 7)
14. Now find another integer x satisfying the conditions of problem 13.
15. What is 232 (mod 30) ?
16. What is 9219080 (mod 19081) ? How about 9219092 ?
(end)
4