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
Homework 3 (Solutions)
(1) We get
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
its prime factorization. Then we get
(i + 1).
Since 3 is prime we get that i + 1 = 1 for all but one of
(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!
when it is divided by
(Hint: Most of the terms are congruent
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
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
(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
(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
(2) For a given odd prime p, how many positiv
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
(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
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