Homework 3 (Solutions)
(1) We get
84
=
22 3 7
846 = 2 3 141
111 = 3 37.
(2) Let n be a number with exactly three divisors. Let n = p1 p2 pk be
1
2
k
its prime factorization. Then we get
3=
(i + 1).
Since 3 is prime we get that i + 1 = 1 for all but one of
Homework 2
(Due: Thursday Feb 9th)
(1) Calculate (n 1)! mod n for n cfw_2, 3, 4, 5, 6, 7, 8, 9, 10, 11.
(2) Find the remainder of
1! + 2! + 3! + + 100!
(3)
(4)
(5)
(6)
when it is divided by
(a) 2
(b) 7
(c) 12
(d) 25
(Hint: Most of the terms are congruent
Homework 8
(1)
(2)
(3)
(4)
Show that there is no positive integer n such that (n) = 14.
Find all positive integers n such that (n)|n.
Show that if m and n are positive integers with m|n, then (m)|(n).
An integer n is perfect if (n) = n. For example
(6) =
Homework 10 Solutions
(1) Note that x is a primitive root if it has order 6 modulo 7. To make sure
x has order 6, it suces to check that does not have order dividing 3 or
2. So, we need to check if x3 and x2 are equivalent to 1 modulo 7. The
following tab
Homework 1
(Due: Thursday Jan 19th)
(1) Assume that there is a surjective map : N S. Show that the map
: S N given by (s) = min 1 (s) is injective.
(2) Assume that : S N is injective. Show that there is a surjective map
: N S. Conclude that subset of a
Homework 2
(Due: Thursday Jan 26th)
Recall that we say a and b are coprime to each other if their greatest common
divisor is 1.
(1) For a given prime p, how many positive integers less than p are coprime to
p?
(2) For a given odd prime p, how many positiv
This is a quick example of Hensels lemma.
Say we want to solve the equation f (x) = x3 + 8x 5 0 (mod p3 ) where p = 5.
We can check that f (0) 0 (mod p). Also f (0) 3 (mod p).
Here is one way of doing this: Let x1 = 0. Let x2 = 0 + pt (since we know
x2 x1
Homework 5 - Solutions
(1) Given a, c, and m, the linear congruence ax c (mod m) is satised if and
only if there is an integer y such that ax + my = c.
Note that the solutions to ax + my = c can be solved using Euclids
algorithm. In some of the easier cas
Homework 7
(1) Find the last digit of the decimal expansion of 71000 .
(2) Show that if a is an integer such that a is not divisible by 3 or such that a
is divisible by 9, then a7 a (mod 63).
(3) Show that a(b) +b(a) 1 (mod ab), if a and b are relatively
Homework 6 - Solutions
(1) Let f (x) = x3 + 8x2 x 1.
(a) Checking all congruences we get that x 4, 5 (mod 11).
(b) We apply Hensels lemma to the two roots x 4 and 5 (mod 11).
x 4 (mod 11) First note that f (4) = 111 1 (mod 11). Therefore, a = 1 is
an inve
Homework 5: Due Nov 29th (Thursday)
(1) Evaluate the limits (note i, j, and k are the three basis vectors in R3 ):
(a) lim (t2 , 4t, 1/t)
t3
(b) lim (e2t i + ln(t + 1)j + 4k).
t0
r(t)
where r(t) = (sin(t), 1 cos(t), 2t).
t
Compute the following:
(a) r (t)
(1) (a) Let
(b) Let
(c) Let
(2) Suppose
Sample midterm 1
an = n n. What is a5 ?
br = 2ar . What is b2 ?
ck = ak ak+1 . Find a formula for ck .
that lim an = 4 and lim bn = 2. Determine
2
n
n
(a) lim an /bn
n
(b) lim sin(b2 )
n
n
(c) lim an2
n
(3) Let a0 =
Sample nal
(1) True or False
(a) There are convergent sequences that are not monotonic.
(b) If lim an = 10 then there is an integer N such that 9.9 < aN < 10.1.
n
(c) There is a sequence an so that a1 = a2 = a3 = = a10 = 1, but
lim an = 0.
n
(d) If 0 < li
Sample midterm 2
(1) The following sequences are convergent. Find their limits.
(a) an = nn+1 .
2
(b) an = n1/n .
(c) a1 = 1, and an+1 = 2 + an for n > 1.
(2) Which of these series converge and why.
n
(a)
log
n+1
n=1
(b)
n
3n
n=1
(c)
1
n(log n + 1)
n=1
1
Homework 5: Due Nov 20th (Tuesday)
x2n
Show that F (x) has innite radius of convergence.
2n n!
n=0
(2) Find the power series for f (x) = x log(1 + x ).
2
1
(3) Find the rst three terms of the Taylor series of f (x) = 1+tan x around 0.
(4) Evaluate
2n
(1)
Homework 5: Due Nov 1st (Thursday)
(1) Find the radius of convergence for
xn
.
2n
n=0
(2) Find the radius of convergence for
x2n
.
n!
n=0
nxn1 .
(3) Find the closed form for the function
n=0
(4) Find the power series expansion for the following functions:
Homework 4: Due Oct 18th (Thursday)
(1) Find the sum of the series i=1 i1 correct to 3 decimal places.
5
1
(2) How many terms of the series n=1 n log(n)2 would you need to add to fnd
its sum to within 0.01?
(3) Find the value of p for which the series n=1
Homework 2: Due Sept 25th
n
(1) Let b1 = 1 and bn+1 = b2 + b1 for all n > 1. Assuming that (bn ) is a
n
convergent sequence, nd its limit.
(2) Let an = (2n + 3n )1/n . Prove that
3 an 21/n 3
for all positive integers n. Calculate lim an .
n
(3) Find a for
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Homework 1: Due Sept 18th
What is a5 for the sequence an = n2 n?
Let a1 = 1 and an = 1 + a1 + a2 + + an1 for n > 1. Calculate the rst
four terms of the sequence (an ).
Let b1 = 2, b2 = 3 and bn = 2bn1 + bn2 for n > 2. Calcu
Homework 4: Due Oct 9th
(1) Which of the following series converge and why?
n1
(a)
n=1 n n
(b)
n=1
n=1
n=1
n=1
n=1
5n
6n +n
(sin n)2
n4 +n
n
en
1
(n!)2
1
sin n
(c)
(d)
(e)
(f)
(2) Using the integral test show that
1
n=2 n log n
diverges.
Rought Solutions
Homework 5: Due Oct 25th (Thursday)
(1) Show that the following series convergent. How many terms of the series
do we need to add in order to nd the sum to the indicated accuracy?
(1)n
(a)
( error less than 0.00005)
n=1 n6
(1)n
(b)
n=1 n5n ( error less
Homework 3: Due Oct 2nd
(1) Rewrite the following series using the Sigma notation.
1
1
(a) 3 + 1 + 27 +
9
1
1
1
1
(b) 1 + 4 + 9 + 16 + 25 +
(1)n
(2) It is known that n=1 n converges. Find
(1)n
(1)n
.
n
n
n=1
n=3
(3) Prove that the following series dive