Question

2. prove that n′ = τ b − κt.

Answer #1

Define the relation τ on Z by a τ b if and only if
there exists x ∈ {1, 4, 16} such that ax ≡ b (mod 63).
(a) Prove that τ is an equivalence relation.
(b) Prove that there exists an integer n with 1 ≤ n ≤ 62 such
that the equivalence class of n is {m ∈ Z | m ≡ n (mod 63)}.

a. Prove for all σ, τ ∈ Sn that στσ−1 τ −1 ∈ An.
b. Let p and q be distinct odd primes. Prove that
Zxpq is not a cyclic group.

Define the relation τ on Z by aτ b if and only if there exists x
∈ {1,4,16} such that
ax ≡ b (mod 63).
(a) Prove that τ is an equivalence relation.
(b) Prove that there exists an integer n with 1 ≤ n ≤ 62 such
that the equivalence class of n is{m ∈ Z | m ≡ n (mod 63)}.

1. ∀n ∈ Z, prove that if ∃a, b ∈ Z such that a 2 + b 2 = n, then
n 6≡ 3 (mod 4).

5. (a) Prove that det(AAT ) = (det(A))2.
(b) Suppose that A is an n×n matrix such that AT = −A. (Such an A
is called a skew- symmetric matrix.) If n is odd, prove that det(A)
= 0.

Prove: If n≡3 (mod 8) and n=a^2+b^2+c^2+d^2, then exactly one of
a, b, c, d is even. (Hint: What can each square be modulo 8?)

Q.Let A and B be n × n matrices such that A = A^2,
B = B^2, and AB = BA = 0.
(a) Prove that rank(A + B) = rank(A) + rank(B).
(b) Prove that rank(A) + rank(In − A) = n.

(a) Let N be an even integer, prove that GCD (N + 2, N) = 2.
(b) What’s the GCD (N + 2, N) if N is an odd integer?

Prove that if a|n and b|n and gcd(a,b) = 1 then ab|n.

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

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