D is a common divisor of a and b, and any common divisor of a and b must divide d. Thus, d is a greatest common divisor of a and b, as required.
To prove that any two nonzero elements of a PID R have a greatest common divisor, let a and b be nonzero elements of R.
First, we note that since R is a PID, it is a UFD (unique factorization domain), and so both a and b have unique factorizations into irreducible elements (i.e., primes) up to units and order.
We define the ideal (a, b) generated by a and b as the set of all elements of the form ra + sb, where r and s are arbitrary elements of R. Since R is a PID, (a, b) is a principal ideal, i.e., (a, b) = (d) for some element d in R.
Now, we claim that d is a greatest common divisor of a and b. To see this, note that d divides both a and b, since a and b are both elements of (d). In other words, there exist elements x and y in R such that a = dx and b = dy. Moreover, any common divisor of a and b must also divide d, since if c divides both a and b, then c also divides any element of the form ra + sb in (a, b), and hence c divides d.
Therefore, d is a common divisor of a and b, and any common divisor of a and b must divide d. Thus, d is a greatest common divisor of a and b, as required.
Therefore, we have shown that any two nonzero elements of a PID R have a greatest common divisor.
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Let R be a principal ideal domain (PID), and let a, b be nonzero elements of R. We need to show that a greatest common divisor (gcd) of a and b exists in R.
Let I be the ideal of R generated by a and b. Since R is a PID, I is a principal ideal, say I = (d) for some element d of R. We claim that d is a gcd of a and b.
First, we show that d is a common divisor of a and b. Since a and b are both in I, they are both multiples of d. Specifically, a = md and b = nd for some elements m, n of R. Therefore, d divides both a and b, and so d is a common divisor of a and b.
Next, we show that d is a greatest common divisor of a and b. Suppose c is another common divisor of a and b. Then c is also a multiple of d, since d generates the ideal (d) containing a and b. Specifically, c = kd for some element k of R. We need to show that d divides c, which would imply that d is a common divisor of a and b that is greater than or equal to c.
Since c is a common divisor of a and b, we have a = xc and b = yc for some elements x, y of R. Substituting c = kd, we obtain a = xkd and b = ykd. Since d is a generator of the ideal (d), it follows that d divides xk and yk. Since R is a domain (meaning that it has no zero divisors), it follows that d divides x and y individually. Therefore, a = xd' and b = yd' for some element d' of R, where d' = xd/gcd(x,y) = yd/gcd(x,y) is another common divisor of a and b. Since gcd(x,y) is a divisor of both x and y, it follows that gcd(x,y) divides d', and therefore d divides d'. This completes the proof that d is a greatest common divisor of a and b.
Therefore, we have shown that any two nonzero elements of R have a greatest common divisor.
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!BRAINLIST! please help me with this QUICK!
Answer:
Corresponding
Co-interior
None
Alternate exterior
Alternate interior
Corresponding
Use | = PRT to solve.
(time is in years)
P = $1,200
I = $99
T= .75 years
Find R.
Can someone help? :)
Answer:
Step-by-step explanation:
PRT = I
R = I/PT
R = 99/(1200)(.75)
R = 99/900
R = 11%
Hi,
So the answer is 11%
How many solutions does this equation have?
4v = 6v − 2v
no solution
one solution
Infinitely many solutions
Answer:
Infinitely many solutions.
Step-by-step explanation:
4v=6v-2v Combine like terms.
4v=4v These are already equal without diving, so there are infinitely many solutions.
Hope this helps and have a nice day ❤
Answer:
Infinitely many solutions
Step-by-step explanation:
4v=6v-2v
4v=4v
when two expressions on opposite sides of an equals sign are equivalent, that means there are infinitely many solutions
P.S. Please mark me Brainliest if it is correct.
hii please help i’ll give brainliest if you give a correct answer!
Answer:
8 and 1/4 cups sugar is your answer. In 5 and 1/3 cups is 8 and 1/4 cups sugar.
Step-by-step explanation:
Brainly please hope this is correct please tell me if I wrong.
Write the expression in expanded form (1 + 3b)(6)
Answer:
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Identity:
An identity is an equality which is true for all values of a variable in the equality.
(a + b)³ = a³+ b³+ 3ab(a + b)
In an identity the right hand side expression is called expanded form of the left hand side expression.
---------------------------------------------------------------------------------------------------
Solution:
(i) (2x + 1)³
Using identity
(a + b)³ = a³+ b³+ 3ab(a + b)
(2x + 1)³
= (2x)³ + 1³ + (3×2x×1)(2x + 1)
= 8x³+ 1 + 6x(2x + 1)
= 8x³ + 1 + 12x² + 6x
(ii) (2a – 3b)³
Using identity,
(a – b)³ = a³–b³ – 3ab(a – b)
(2a – 3b)³ = (2a)³– (3b)³ – (3×2a×3b)(2a – 3b)
=8a³–27b³–18ab(2a –3b)
= 8a³–27b³–36a²b + 54ab²
(iii) [3x/2 + 1]³
Using identity,
(a + b)³ = a³+ b³+ 3ab(a + b)
[3x/2 +1]³
=(3x/2)³+1³+ (3×(3x/2)×1)(3x/2+ 1)
=27x³/8+1+9/2x×(3x/2+1)
= 27x³/8 + 1 + 27/4 x² + 9/2x
= (27/8)x³ + (27/4) x² + 9/2 x + 1
(iv) [x–2/3 y]³
Using identity,
(a - b)³=a³-b³-3ab(a-b)
(X+ 2/3y)³
= (x)³–(2/3 y)³– (3×x×2/3 y)(x – 2/3 y)
= x³– 8y³/27–2xy(x – 2/3 y)
= x³– (8/27)y³–2x²y+ 4/3xy²
=========================================================
Hope this will help you....
Step-by-step explanation:
Given the sequence 24, 12, 6, 3......., where x = 1, 2, 3, 4, .........What is the first term? 24 12 6 3
Answer:
24
Step-by-step explanation:
The first term of any sequence is the term of the sequence that comes first.
Given the sequence of numbers 24, 12, 6, 3...
We can see that the number that comes first is 24. Hence the first term of the given sequence will be 24
i need help with this anyone pleaseee
Answer:
41 dollars
Step-by-step explanation:
3r+2.5p is the cost of roses and petunias, just number of roses as r and petunias as p:
3(7) + 2.5(8)=
21+20=41 dollars
Но
How much does each bead necklace cost?
You can buy a 35-pound box of beads for $43.95.
A 35-pound box of beads contains 30 bags of 12
necklaces each.
Step 1: How many necklaces are in a box?
B, the correct answer is B
The price of an item has been reduced by 95% the original price was 87$
Answer:
Your answer
Price after reducing is 4.35$
Mark it as Brainlist. Follow me for more answer.
Step-by-step explanation:
PLS HELP!!
Solve for an angle in right triangles
B = ???
round your answer to the nearest hundredth
Answer:
100
Step-by-step explanation:
Triangle B C D is shrunk to form triangle B prime C prime D prime.
Identify the type of transformation that maps triangle BCD onto triangle B’C’D’.
The transformation is a
Answer:
dilation.
Step-by-step explanation:
Answer:
dilation
Step-by-step explanation:
if 6n = 0, what does n equal?
Answer:
6n=0
n=0/6
n=0
hope it helps.
Answer:
n=0
Step-by-step explanation:
6n is the same of 6×n
∴ 6×0 = 0
A right circular cylinder is 3m high and the circumference of its base is 22m. Find its curved surface area.
Answer:
height of cylinder =3m
circumference =22m
curved surface area=2×pi×radius×height
curved surface area =22m×3m
curved surface area =66m²
At a sale, coats were sold for 82% of their original price. If the coats originally cost $200 each, how much did a coat cost on sale?
164
Step-by-step explanation:
The solution is, a coat cost on sale = $164.
What is percentage?A percentage is a number or ratio that can be expressed as a fraction of 100. A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number; it has no unit of measurement.
here, we have,
At a sale, coats were sold for 82% of their original price.
If the coats originally cost $200 each,
so, a coat cost on sale = $200* 82%
=$164
Hence, The solution is, a coat cost on sale = $164.
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Sam complete 3 math problems in 8 minutes. How long will it take to do 15 problems?
Answer:
40 minutes
Step-by-step explanation:
Math Problems 3 15
---------------------- ----- = ------
Minutes 8 x
3x = 15 * 8
x = 15 * 8 / 3
x = 120 / 3
x = 40
40 Minutes
HELP I NEED HELP ASAP
Answer:
d.
Step-by-step explanation:
Answer:
option c I guess - "It increases exponentially "
Find an equation for line L, M, N, and P *
4 _ (3 _ 2) _ 6 _ 1 = 14 Please help ill give you brainly!
angle (the question mark), (Write the
number only)
?
77
124°
47. my answer needs to be at least 20 characters long there u go
Answer:
49
Step-by-step explanation:
77+49 = 124
Based on the side lengths alone, could the triangles be similar? 16.0 cm R 6.4 cm 3.0 cm 15.0 cm S T 6.0 cm X 7.5 cm U
No, the side are not proportional.
Yes, the sides are in the ratio 2:5.
yes, the sides are in the ratio 4:5.
Yes, the sides are in the ratio 1:5.
Answer:
Yes, the sides are in the ration 2:5!
Step-by-step explanation:
Hope this helps.
Answer:
The correct answer is B. Yes, the sides are in the ratio 2:5.
Step-by-step explanation:
Hope this helps, and have a wonderful day! :)
Ten people put 63 boxes on a red truck and 38 boxes on a blue track About how many boxes did they load in all
Answer:
63+38=101
Step-by-step explanation:
please answer correctly and quickly
Answer:
#6:10
Step-by-step explanation:
Answer:
#9 is x=14
#7 is C 138
#6 is x=10
#5 is p=40
What is 25% of 160?
Instructions
100 of 100 words remaining
BUI EEE Tv
b. What is 39 of 2007
Instructions
100 watt
Done and Review
2021
Answer:
I have no idea what ur saying
Step-by-step explanation:
b-3/15=1 What does b equal??
Answer:
1 3/15
Step-by-step explanation:
All you have to do is add 3/15 to both sides of the equation:
b - 3/15 = 1
b = 1 3/15
HELP AND WILL GIVE BRAINLIEST!!! Solve for x for the image below
Answer:
umm x=30
Step-by-step explanation:
Answer:
x=22
Step-by-step explanation:
All angles in a triangle add up to 180 degree.
180-(6x-7) = the supplementary angle inside the upper left corner.
180-(6x-7)+2x+(103-x) = 180
180-6x+7+2x+103-x = 180
reduce
-5x = -110
-5x/-5 = -110/-5
x=22
you can check by using the values of x
5/16 + 7/16 (simpfiy it )
Two balls are picked in succession without replacement -4 white balls and 5 green balls let Y be the random representing the number of green balls. Find the value of the random variable Y.
Answer:
If Y represents the number of green balls then it can have values from 0- 5
that is it can have all the value 0,1,2,3,4 and 5.
The probability of getting both balss green without replacement is 0.27778
Step-by-step explanation:
If the balls are not replaced then this is the dependent event .
The probability of getting the first ball is given be = 5/9
where 5 is the number of green balls and 9 is the total number of balls.
The probability of getting the second ball green given that first is also green is = 5/9* 4/8= 20/72= 0.2778
If Y represents the number of green balls then it can have values from 0- 5
that is it can have all the value 0,1,2,3,4 and 5.
0 means there can be no green balls or white balls.
1 means there can be one green ball.
2 means there can be 2 green balls and so on.
In this exercise we have to use the knowledge of probability to calculate the chance of an event to occur, so you have to:
the chance of the event is 0.2778%
So organizing the information given in the statement is:
If the balls are not replaced then this is the dependent event .The probability of getting the first ball is given be = 5/95 is the number of green balls and 9 is the total number of balls.So with this information we can write that:
[tex]5/9* 4/8= 20/72= 0.2778[/tex]
See more about probability at brainly.com/question/795909
At the zoo, the ratio of birds to snakes is 2:3. If there
are 12 snakes at the zoo, how many birds are there?
Show your work! Type your final answer in the box
below.
Submit
Answer:
8 birds
Step-by-step explanation:
Birds to snakes is 2:3. So, then it would be 4:6, 6:9, 8:12 if you use a table.
Answer:
U need to slove the same way as shown above just change the number and solve if still u find any difficulty. then u can follow me I will surely help u out
All The Best
Can someone help me please. I have to show my work.
Answer:
Step-by-step explanation:
1. P = (width + lengh) * 2 = (23 + 10) * 2 = 33 * 2 = 66
A = width * lengh = 23 * 10 = 230
2. width = 7
P = (width + lengh) * 2 = ( 7 + 9) * 2 = 16 * 2 = 32
Answer:
4. area= 230cm^2 perimeter= 66cm
5. width= 7m perimeter= 32m
Step-by-step explanation:
4. a = w*l
23cm*10cm=230cm^2
p= add all sides
23cm+23cm+10cm+10cm=66cm
5. w= a/l
63m^2/9m=7m
p= add all sides
7m+7m+9m+9m=32m
what is the square root of 4.84
Answer:
2.2. Hope it helps :D