Question

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.

Answer #1

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 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 by contradiction that:
If n is an integer greater than 2, then for all integers m, n
does not
divide m or n + m ≠ nm.

In the style of the proof that square root of 2 is irrational,
prove that the square root of 3 is irrational. Remember, we used a
proof by contradiction. You may use the result of Part 1 as a
"Lemma" in your proof.

Prove that if n is a positive integer greater than 1,
then n! + 1 is odd
Prove that if a, b, c are integers such that a2 + b2 =
c2, then at least one of a, b, or c is even.

Let n be an integer, with n ≥ 2. Prove by contradiction that if
n is not a prime number, then n is divisible by an integer x with 1
< x ≤√n.
[Note: An integer m is divisible by another integer n if there
exists a third integer k such that m = nk. This is just a formal
way of saying that m is divisible by n if m n is an integer.]

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!)

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