Let n be a positive integer. Prove that if n is composite, then
n has a...
Let n be a positive integer. Prove that if n is composite, then
n has a prime factor less than or equal to sqrt(n) . (Hint: first
show that n has a factor less than or equal to sqrt(n) )
(a) Let a > 1 be an integer. Prove that any composite divisor
of a − 1 is a pseudoprime of base a.
(b) Suppose, for some m, than n divides a^(m − 1) and n ≡ 1 (mod
m). Prove that if n is composite, then n is a pseudoprime of base
a.
(c) Use (b) to give two examples pseudoprimes of base a with a =
2 and a = 3 (hint: take m = 2k to be...
Let G be an abelian group and n a fixed positive integer. Prove
that the following sets are subgroups of G.
(a) P(G, n) = {gn | g ∈ G}.
(b) T(G, n) = {g ∈ G | gn = 1}.
(c) Compute P(G, 2) and T(G, 2) if G = C8 ×
C2.
(d) Prove that T(G, 2) is not a subgroup of G = Dn
for n ≥ 3 (i.e the statement above is false when G is...
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
Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2)
− (1/3j + 1)). a) Compute the value of S(1), S(2), S(3), and S(4).
b) Make a conjecture that gives a closed form (i.e., not a
summation) formula for the value of S(n). c) Use induction to prove
your conjecture is correct.