Question

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

Answer #1

Prove that for every positive integer n, there exists a multiple
of n that has for its digits only 0s and 1s.

Prove that for every positive integer n, there exists an
irreducible polynomial of degree n in Q[x].

(a) Prove or disprove the statement (where n is an integer): If
3n + 2 is even, then n is even.
(b) Prove or disprove the statement: For irrational numbers x
and y, the product xy is irrational.

Let a be prime and b be a positive integer. Prove/disprove, that
if a divides b^2 then a divides b.

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

Prove or disprove the following statement: 2^(n+k) is an element
of O(2^n) for all constant integer values of k>0.

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

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!)

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.)

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).

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