Question

Suppose p is a positive prime integer and k is an integer satisfying 1 ≤ k ≤ p − 1. Prove that p divides p!/ (k! (p-k)!).

Answer #1

Above rule applicable of all types of natural number (not just for the prime numbers)

Prove that for any integer a, k and prime p, the following three
statements are all equivalent: p divides a, p divides a^k, and p^k
divides a^k.

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

Let A be a 2 × 2 matrix satisfying A^k = 0 for some positive
integer k. Show that A^2 = 0.

If p = 2k − 1 is prime, show that k is an odd integer or k =
2.
Hint: Use the difference of squares 22m − 1 = (2m − 1)(2m +
1).

Suppose for each positive integer n, an is an integer such that
a1 = 1 and ak = 2ak−1 + 1 for each integer k ≥ 2. Guess a simple
expression involving n that evaluates an for each positive integer
n. Prove that your guess works for each n ≥ 1.
Suppose for each positive integer n, an is an integer such that
a1 = 7 and ak = 2ak−1 + 1 for each integer k ≥ 2. Guess a...

Let p be an odd prime and let a be an odd integer with p not
divisible by a. Suppose that p = 4a + n2 for some
integer n. Prove that the Legendre symbol (a/p) equals 1.

Formal Proof: Let p be a prime and let a be an
integer. Assume p ∤ a. Prove gcd(a, p) = 1.

4. Prove that if p is a prime number greater than 3, then p is
of the form 3k + 1 or 3k + 2.
5. Prove that if p is a prime number, then n √p is irrational
for every integer n ≥ 2.
6. Prove or disprove that 3 is the only prime number of the form
n2 −1.
7. Prove that if a is a positive integer of the form 3n+2, then
at least one prime divisor...

Activity 6.6.
(a)
A positive integer that is greater than 11 and not
prime is called composite.
Write a technical definition for the concept of composite number
with a similar level of detail as in the “more complete” definition
of prime number.
Note.
A number is called prime if its only divisors are 1 and
itself.
This definition has some hidden parts: a more complete
definition would be as follows.
A number is called prime if
it is an integer,...

let x be a discrete random variable with positive integer
outputs.
show that P(x=k) = P( x> k-1)- P( X>k) for any positive
integer k.
assume that for all k>=1 we have P(x>k)=q^k. use (a) to
show that x is a geometric random variable.

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