Let p be an integer other than 0, ±1.
(a) Prove that p is prime if and only if it has the property
that whenever r and s are integers such that p = rs, then either r
= ±1 or s = ±1.
(b) Prove that p is prime if and only if it has the property
that whenever b and c are integers such that p | bc, then either p
| b or p | c.
Please prove
1. Every positive integer is a product of prime numbers.
2. If a and b are relatively prime, and a|bc, then a|c.
3. The division algorithm for F[x]. Just the existence part
only, not the uniqueness part
Let p be an odd prime and a be any integer which is not
congruent to 0 modulo p.
Prove that the congruence x 2 ≡ −a 2 (mod p) has solutions if
and only if p ≡ 1 (mod 4).
Hint: Naturally, you may build your proof on the fact that the
statement to be proved is valid for the case a = 1.
Suppose that A is a triangle with integer sides and integer
area. Prove that the semiperimeter of A cannot be a prime
number.
(Hint: Suppose that a natrual number x is a perfect square, and
suppose that p is a prime number that divides x. Explain why it
must be the case that p divides x an even number of times)
Let p be an odd prime.
(a) (*) Prove that there is a primitive root
modulo p2 . (Hint: Use that if a, b have orders n, m,
with gcd(n, m) = 1, then ab has order nm.)
(b) Prove that for any n, there is a primitive
root modulo pn.
(c) Explicitly find a primitive root modulo
125.
Please do all parts.
Thank you in advance
8.Let a and b be integers and d a positive
integer.
(a) Prove that if d divides a and d divides b, then d divides both
a + b and a − b.
(b) Is the converse of the above true? If so, prove it. If not,
give a specific example of a, b, d showing
that the converse is false.
9. Let a, b, c, m, n be integers. Prove that if a divides each of b
and c,...
Let t be a positive integer. Prove that, if there exists a
Steiner triple system of index 1 having v varieties, then there
exists a Steiner triple system having v^t varieties