Question

Let S = {x ∈ Z : −60 ≤ x ≤ 59}.

(a) Which integers are both in S and 6Z?

(b) Which integers in S have 1 as the remainder when divided by 6?

(c) Which integers in S are also in −1 + 6Z?

(d) Which integers satisfy n ≡ 3 mod 6?

Answer #1

Let Z be the integers.
(a) Let C1 = {(a, a) | a ∈ Z}. Prove that
C1 is a subgroup of Z × Z.
(b) Let n ≥ 2 be an integer, and let Cn = {(a, b) | a
≡ b( mod n)}. Prove that Cn is a subgroup of Z × Z.
(c) Prove that every proper subgroup of Z × Z that contains
C1 has the form Cn for some positive integer
n.

Let a, b, c, m be integers with m > 0. Prove the following:
(a) ”a ≡ 0 (mod 2) if and only if a is even” and ”a ≡ 1 (mod 2) if
and only if a is odd”. (b) a ≡ b (mod m) if and only if a − b ≡ 0
(mod m) (c) a ≡ b (mod m) if and only if (a mod m) = (b mod m).
Recall from Definition 8.10 that (a...

Give an example of three positive integers m, n, and r, and
three integers a, b, and c such that the GCD of m, n, and r is 1,
but there is no simultaneous solution to
x ≡ a (mod m)
x ≡ b (mod n)
x ≡ c (mod r).
Remark: This is to highlight the necessity of “relatively prime”
in the hypothesis of the Chinese Remainder Theorem.

Given non-zero integers a, b ∈ Z, let X := {ra + sb | r, s ∈ Z
and ra + sb > 0}. Then: GCD(a, b) is the least element in X.

Suppose n and m are integers. Let H = {sm+tn|s ∈ Z and t ∈
Z}.
Prove that H is a cyclic subgroup of Z.
......................
Please help with clear steps that H is a cyclic subgroup of
Z

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

Let f ∈ Z[x] be a nonconstant polynomial. Prove that the set S =
{p prime: there exist infinitely many positive integers n such that
p | f(n)} is infinite.

Let X be a subset of the integers from 1 to 1997 such that
|X|≥34. Show that there exists distinct a,b,c∈X and distinct
x,y,z∈X such that a+b+c=x+y+z and {a,b,c}≠{x,y,z}.

Let N denote the set of positive integers, and let x be a number
which does not belong to N. Give an explicit bijection f : N ∪ x →
N.

Answer the following question:
1.
a. Use an affine cipher x 7→ 3x + 1 (mod 26) to encode
“Baltimore”.
b. Let a and b be integers. What does it mean to say a divides
b? Provide a precise definition and include the proper
notation.
c. Let a, b, c, and n be integers with n 6= 0. Suppose that a ≡
b (mod n) and b ≡ c (mod n). Prove that a ≡ c (mod n).
d. Use...

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