Let m, n be natural numbers such that their greatest common
divisor gcd(m, n) = 1. Prove that there is a natural number k such
that n divides ((m^k) − 1).
a) Compute the indicated quantity.
P(A | B) = .1, P(B) = .4. Find P(A ∩ B).
P(A ∩ B) =
b)Compute the indicated quantity.
P(A) = .1, P(B) = .2. A and B are independent. Find P(A ∩
B).
P(A ∩ B) =
c)Find the conditional probability of the indicated event when
two fair dice (one red and one green) are rolled. HINT [See Example
1.]
The red one is 1, given that the sum is 7.
Suppose we define a relation on the set of natural numbers as
follows. Two numbers are related iff they leave the same remainder
when divided by 5. Is it an equivalence relation? If yes, prove it
and write the equivalence classes. If no, give formal
justification.
Required:
1. For direct materials:
a. Compute the price and quantity variances.
b. The materials were purchased from a new supplier who is
anxious to enter into a long-term purchase contract. Would you
recommend that the company sign the contract?
2. For direct labor:
a. Compute the rate and efficiency variances.
b. In the past, the 20 technicians employed in the production of
Fludex consisted of 7 senior technicians and 13 assistants. During
November, the company experimented with fewer senior...
For an integer k, define f(k) = gcd(11k + 1, 7k + 3).
(a) Compute R = {f(k): k ∈ Z}.
(b) For each n ∈ R, find a set Dn such that, for every
integer k, f(k) = n if and only if k ∈ Dn.
Is there any solution without using the 'mod' for
b?
Show the following identities for a, b, c ∈ N.
(a) gcd(ca, cb) = c gcd(a, b) Hint: To show that two integers x,
y ∈ Z are equal you can show that both x | y and y | x which
implies x = y or x = −y. Thus, if both x and y have the same sign,
they must be equal.
(b) lcm(ca, cb) = c lcm(a, b)
(c) ab = lcm(a, b) gcd(a, b) Hint: Consider...
(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then
a∣c.
(B) Let p ≥ 2. Prove that if 2p−1 is prime, then p
must also be prime.
(Abstract Algebra)
The number 73 is written as a sum of three natural numbers
73=a+b+c
(the triple (a,b,c) is ordered; e.g., the decompositions
73=19+20+34 and 73=20+34+19 are different.
Also, assume that all the decompositions have equal
probability.)
Given that there exists a triangle with sides a, b, and c, what is
the probability that this triangle is isosceles?