In: Advanced Math

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 first the case that gcd(a, b) = 1 and show that ab = lcm(a, b) in this case. For the general case combine this with (b).

(d) lcm(gcd(a, c), gcd(b, c)) = gcd(lcm(a, b), c) Hint: First treat the special case that gcd(a, b, c) = 1. In this case begin by showing that lcm(gcd(a, c), gcd(b, c)) = gcd(a, c) gcd(b, c). The asserted equality gcd(a, c) gcd(b, c) = gcd(lcm(a, b), c) is then shown by proving that gcd(a, c) gcd(b, c)| gcd(lcm(a, b), c) and gcd(lcm(a, b), c)| gcd(a, c) gcd(b, c). Proceed to show that gcd(a, c)| lcm(a, b) and gcd(a, c)| c, and deduce from this that gcd(a, c)| gcd(lcm(a, b), c); proceed analogously for gcd(b, c). Then argue that gcd(a, c) gcd(b, c)| gcd(lcm(a, b), c) under the present assumption. Conversely, in order to show that gcd(lcm(a, b), c)| gcd(a, c) gcd(b, c), write according to (a) gcd(a, c) gcd(b, c) = gcd(gcd(a, c)b, gcd(a, c)c) = gcd(gcd(ab, bc), gcd(ac, c2 )), and show that gcd(lcm(a, b), c) divides all of ab, bc, ac, and c 2 . Explain from here why gcd(lcm(a, b), c) must divide gcd(a, c) gcd(b, c) then as well. For the general case explain how (a) and (b) can be used to reduce the general assertion to the previously treated special case.

***The only help I really need is with c and d. I just added a and b for context.

Buffer 1: 0.1M CA mixed with 0.1M CB
Buffer 2: 0.01M CA mixed with 0.01M CB
Buffer 3: 0.1M CA mixed with 0.01 M CB
Buffer 4: 0.01M CA mixed with 0.1 M CB
This question is from Buffer Capacity lab. Please give me a
detailed explanation
How would I make comments on the buffer capacity when add HCl
and NaOH to buffer solutions which contain different concentration
of CA (conjugate acid) and CB?

Prove that gcd(a,b) = gcd(a+b,lcm(a,b))

Prove that for arbitrary sets A, B, C the following
identities are true. Note that Euler Diagram is not a proof but can
be useful for you to visualize!
(A∩B)⊆(A∩C)∪(B∩C')
Bonus question:
A∪B∩A'∪C∪A∪B''=
=(A∩B∩C)∪(A∩B'∩C)∪(A'∩B∩C)∪(A'∩B∩C')

(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)

For each of the following reactions identify the conjugate acid
(CA) and conjugate base (CB):
Reaction
1: HCO3-
+ H2O (gives)
H3O+ + CO32-
Reaction 2:
HSO4-
+
HCN (gives) CN- + H2SO4
A.
Reaction 1: CA: H3O+, CB :
CO32-, Reaction 2: CA:
H2SO4 , CB: CN-
B.
Reaction 1: CB: H3O+, CA :
CO32-, Reaction 2: CA:
H2SO4 , CB: CN-
C.
Reaction 1: CA: H3O+, CB :
CO32-, Reaction 2: CB:
H2SO4 , CA: CN-
D.
Reaction 1: CB: H3O+, CA :
CO32-, Reaction 2:...

1. Must be nicely written up AS A PROOF.
a. Show that gcd(m + n, m) = gcd(m, n).
b. If n | k(n + 1), show that n | k.
c. Show that any two consecutive odd integers are relatively
prime.

Write the COMPLETE electron configurations for the following
elements or ions.
a. Ca
b. As3-
c. Ti4+
d. Zn

Thank You
Define the gcd of three integers a, b, c as the largest common
divisor of a, b, c, and denote it by (a, b, c). Show that (a, b, c)
= ((a, b), c) and that (a, b, c) can be expressed as a linear
combination of a, b, c.

Consider the following economy C = 0.85 (Y – T) + Ca ; Ca = 600
– 25 R ; T = 450 + 0.225 Y; IP = 1500 – 30 R ; G =1900; NX = 950 –
0.0625 Y. a) What are the values of the autonomous net export NXa
and the autonomous taxes Ta (hint: see the formulas and compare) b)
Compute the multiplier c) Derive the equation of the autonomous
spending. d) Derive the equation of...

Let G be a group of order mn where gcd(m,n)=1
Let a and b be elements in G such that o(a)=m and 0(b)=n
Prove that G is cyclic if and only if ab=ba

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