Let m, n be natural numbers such that their greatest common
divisor gcd(m, n) = 1. Prove that there is a natural number k such
that n divides ((m^k) − 1).
There are (m − 1)n + 1 people in a room. Show that either there
are m people who mutually do not know each other, or there is a
person who knows at least n others.
i
need a very detailed proof
(Show
your work!)
Let n
> 1. Prove: The sum of the positive integers less than or equal
to n is a divisor of the product of the positive integers less than
or equal to n if and only if n + 1 is composite.
Give a proof by contradiction to show that if two lines l and
m are cut by a transversal in such a way that the alternate
interior angles, x and y, have the same measure, then the lines are
parallel.
Write the "if, then" statement for the proof by contradiction
and the proof.
D4 = {(1),(1, 2, 3, 4),(1, 3)(2, 4),(1, 4, 3, 2),(1,
2)(3, 4),(1, 4)(2, 3),(2, 4),(1, 3)}
M = {(1),(1, 4)(2, 3)}
N = {(1),(1, 4)(2, 3),(1, 3)(2, 4),(1, 2)(3, 4)}
Show that M is a subgroup N; N is a subgroup D4, but
that M is not a subgroup of D4
How to proof:
Matrix A have a size of m×n, and the rank is r. How can we
rigorous proof that the dimension of column space are always equal
to the dimensional of row space?
(I can use many examples to show this work, but how to proof
rigorously?)