Question

1. Let p be any prime number. Let r be any integer such that 0 < r < p−1. Show that there exists a number q such that rq = 1(mod p)

2. Let p1 and p2 be two distinct prime numbers. Let r1 and r2 be such that 0 < r1 < p1 and 0 < r2 < p2. Show that there exists a number x such that x = r1(mod p1)andx = r2(mod p2).

8. Suppose we roll a standard six-sided dice over and over again. What is the expected number of rolls until we got the first pair of consecutive sixs?

Answer #1

: (a) Let p be a prime, and let G be a finite Abelian group.
Show that Gp = {x ∈ G | |x| is a power of p} is a subgroup of G.
(For the identity, remember that 1 = p 0 is a power of p.) (b) Let
p1, . . . , pn be pair-wise distinct primes, and let G be an
Abelian group. Show that Gp1 , . . . , Gpn form direct sum in...

Let p and q be any two distinct prime numbers and define the
relation a R b on integers a,b by: a R b iff b-a is divisible by
both p and q. For this relation R: Show that the equivalence
classes of R correspond to the elements
of ℤpq. That is: [a] = [b] as equivalence
classes of R if and only if [a] = [b] as elements of
ℤpq.
you may use the following lemma: If p is prime...

Let p and q be any two distinct prime numbers and define the
relation a R b on integers a,b by: a R b iff b-a is divisible by
both p and q. For this relation R: Prove that R is an equivalence
relation.
you may use the following lemma: If p is prime and p|mn, then
p|m or p|n

Let p and q be any two distinct prime numbers and define the
relation a R b on integers a,b by: a R b iff b-a is divisible by
both p and q.
I need to prove that:
a) R is an equivalence relation. (which I have)
b) The equivalence classes of R correspond to the elements
of ℤpq. That is: [a] = [b] as equivalence
classes of R if and only if [a] = [b] as elements of
ℤpq
I...

Let p be prime. Show that the equation x^2 is congruent to 1(mod
p) has just two solutions in Zp (the set of integers). We cannot
use groups.

Let E be a field of characteristic p, where p is a prime number.
Show that for all x, y that are elements of E, we have (x + y)^p
=x^p + y^p, and hence by induction, (x + y)^p^n = x^p^n + y^p^n
.

Let G be a group and let p be a prime number such that
pg = 0 for every element g ∈ G.
a. If
G is commutative under multiplication, show that the mapping
f : G → G
f(x) =
xp
is a homomorphism
b. If G is
an Abelian group under addition, show that the mapping
f : G → G
f(x) = xpis a homomorphism.

Definition: Let p be a prime and 0 < n then the p-exponent of
n, denoted ε(n, p) is the largest number k such that pk | n.
Note: for p does not divide n we have ε(n,p) = 0
Notation: Let n ∈ N+ we denote the set {p : p is prime and p |
n} by Pr(n). Observe that Pr(n) ⊆ {2, 3, . . . n} so that Pr(n) is
finite.
Problem: Let a, b be...

Let p be a prime and m an integer. Suppose that the polynomial
f(x) = x^4+mx+p is reducible over Q. Show that if f(x) has no zeros
in Q, then p = 3.

Suppose you roll two dice and count the number of dots showing.
Let A = "the sum of the dots showing on the two rolls is a prime
number."
(a) Let B = "The first toss showed an even number of dots." Are
A and B independent? Show your work.
(b) Let C = "the second toss was greater than the first." Are A
and C independent? Show your work.
(c) Suppose we pick a value ?k for 1≤?≤61≤k≤6 and...

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