Question

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.

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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.

Let n be an integer. Prove that if n is a perfect square (see
below for the definition) then n + 2 is not a perfect square. (Use
contradiction) Definition : An integer n is a perfect square if
there is an integer b such that a = b 2 . Example of perfect
squares are : 1 = (1)2 , 4 = 22 , 9 = 32 , 16, · ·
Use Contradiction proof method

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.

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 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 that the square
root of 17 is irrational. Subsequently, prove that n times the
square root of 17 is irrational too, for any natural number n.
use the following
lemma: Let p be a prime number; if p | a2 then p | a as
well. Indicate in your proof the step(s) for which you invoke this
lemma. Check for yourself (but you don’t have to include it in your
worked solutions) that this need not be true if...

Prove that the square root of 17 is irrational. Subsequently,
prove that n times the square root of 17 is irrational too, for any
natural number n.
use the following lemma: Let p be a prime number; if p |
a2 then p | a as well. Indicate in your proof the
step(s) for which you invoke this lemma. Check for yourself (but
you don’t have to include it in your worked solutions) that this
need not be true if...

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

Suppose that Y has a gamma distribution with α = n/2 for some
positive integer n and β equal to some specified value. Use the
method of moment-generating functions to show that W = 2Y/β has a
χ2 distribution with n degrees of freedom. (Please show all work,
proof used, and logic to justify the answer. Thank, you)

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