Question

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

Answer #1

3. Prove by contrapositive: Let n ∈ N. If n^3−5n−10>0,then n
≥ 3.
4. Prove: Letx∈Z. Then5x−11 is even if and only if x is odd.
4. Prove: Letx∈Z. Then 5x−11 is even if and only if x is
odd.

Prove or disprove that 3|(n^3 − n) for every positive integer
n.

Use mathematical induction to prove that for each integer n ≥ 4,
5n ≥ 2 2n+1 + 100.

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 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!)

a) Prove: If n is the square of some integer, then n /≡ 3 (mod
4). (/≡ means not congruent to)
b) Prove: No integer in the sequence 11, 111, 1111, 11111,
111111, . . . is the square of an integer.

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.

Prove that 2n < n! for every integer n ≥ 4.

Show 2 different solutions to the task.
Prove that for every integer n (...-3, -2, -1, 0, 1, 2, 3,
4...), the expression n2 + n will always be even.

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