Question

Prove that n is prime iff every linear equation ax ≡ b mod n, with a ≠ 0 mod n, has a unique solution x mod n.

Answer #1

Prove that for a square n ×n matrix A, Ax = b (1) has one and
only one solution if and only if A is invertible; i.e., that there
exists a matrix n ×n matrix B such that AB = I = B A.
NOTE 01: The statement or theorem is of the form P iff Q, where
P is the statement “Equation (1) has a unique solution” and Q is
the statement “The matrix A is invertible”. This means...

For a and b relatively prime, prove that the
largest k for which ax + by = k with x and y non-negative integers
has no solution is k = ab - a - b.

prove that n> 1 and (n-1)!congruence -1(mod n) then n is
prime

Prove that for every integer n, 30 | n iff 5 | n and 6 | n. This
problem is similar to examples and exercises in Section 3.4 of your
SNHU MAT299 textbook

Prove that if ? ≡ ? (mod n) and ? ≡ ? (mod n), then ? ≡ ? (mod
n). This proves that congruence mod n is transitive.
and
: Prove that if ? ≡ ? (mod n) and ? ≡ ? (mod n), then
a) ? + ? ≡ ? + ? (mod n)
b) ?? ≡ ?? (mod n)

Let p be a prime that is congruent to 3 mod 4. Prove that there
is no solution to the congruence x2≡−1 modp. (Hint: what would be
the order of x?)

(§2.1) Let a,b,p,n ∈Z with n > 1.
(a) Prove or disprove: If ab ≡ 0 (mod n), then a ≡ 0 (mod n) or
b ≡ 0 (mod n).
(b) Prove or disprove: Suppose p is a positive prime. If ab ≡ 0
(mod p), then a ≡ 0 (mod p) or b ≡ 0 (mod p).

(a) Let a,b,c be elements of a field F. Prove that if a not= 0,
then the equation ax+b=c has a unique solution.
(b) If R is a commutative ring and x1,x2,...,xn are independent
variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is
isomorphic to R[x1,x2,...,xn] for any permutation σ of the set
{1,2,...,n}

(b) If n is an arbitrary element of Z, prove directly that n is
even iff n + 1 is odd. iff is read as “if and only if”

Let gcd(m1,m2) = 1. Prove that a ≡ b (mod m1) and a ≡ b (mod m2)
if and only if (meaning prove both ways) a ≡ b (mod m1m2). Hint: If
a | bc and a is relatively prime to to b then a | c.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 22 minutes ago

asked 41 minutes ago

asked 41 minutes ago

asked 42 minutes ago

asked 53 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago