Question

Prove the following theorem: For every integer n, there is an even integer k such that

n ≤ k+1 < n + 2.

Your proof must be **succinct** and cannot contain
more than **60** words, with equations or inequalities
counting as one word. Type your proof into the answer box. If you
need to use the less than or equal symbol, you can type it as <=
or ≤, but the proof can be completed without it.

Answer #1

Prove the following theorem: For every integer n, there is an
even integer k such that
n ≤ k+1 < n + 2.
Your proof must be succinct and cannot contain more than 60
words, with equations or inequalities counting as one word. Type
your proof into the answer box. If you need to use the less than or
equal symbol, you can type it as <= or ≤, but the proof can be
completed without it.

Discrete Math
6. Prove that for all positive integer n, there exists an even
positive integer k such that
n < k + 3 ≤ n + 2
. (You can use that facts without proof that even plus even is
even or/and even plus odd is odd.)

Prove that there is no positive integer n so that 25 < n^2
< 36. Prove this by directly proving the negation.Your proof
must only use integers, inequalities and elementary logic. You may
use that inequalities are preserved by adding a number on both
sides,or by multiplying both sides by a positive number. You cannot
use the square root function. Do not write a proof by
contradiction.

Prove that there is no positive integer n so that 25 < n2
< 36. Prove this by directly proving the negation. Your proof
must only use integers, inequalities and elementary logic. You may
use that inequalities are preserved by adding a number on both
sides, or by multiplying both sides by a positive number. You
cannot use the square root function. Do not write a proof by
contradiction.

Let n be an even integer. Prove that Dn/Z(Dn) is isomorphic to
D(n/2). Prove this using the First Isomorphism Theorem

Suppose n ≥ 3 is an integer. Prove that in Sn every
even permutation is a product of cycles of length 3.
Hint: (a, b)(b, c) = (a, b, c) and (a, b)(c, d) = (a, b, c)(b,
c, d).

Prove that every integer of the form 5n + 3 for n ∈ Z, n ≥ 1,
cannot be a perfect square

“For every nonnegative integer n, (8n – 3n) is a multiple of
5.”
(That is, “For every n≥0, (8n – 3n) = 5m, for some m∈Z.” )
State what must be proved in the basis
step.
Prove the basis step.
State the conditional expression that must be proven in the
inductive step.
State what is assumed true in the inductive hypothesis.
For this problem, you do not have to complete the inductive step
proof. However, assuming the inductive step proof...

Prove that a positive integer n, n > 1, is a perfect square
if and only if when we write
n =
P1e1P2e2...
Prer
with each Pi prime and p1 < ... <
pr, every exponent ei is even. (Hint: use the
Fundamental Theorem of Arithmetic!)

Prove let n be an integer. Then the following are
equivalent.
1. n is an even integer.
2.n=2a+2 for some integer a
3.n=2b-2 for some integer b
4.n=2c+144 for some integer c
5. n=2d+10 for some integer d

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