Please help as soon as possible <br>it's due tomorrow

13. The least common multiple of two non-zero integers a and b is the unique positive integer m such that (i) m is a common mult

Vlad [161]

Answer:

[a,b] divides n

Step-by-step explanation:

Let us denote the least common multiple of a and b [a,b]=m.

We want to prove that m divides n, where n is a multiple of a and b.

We suppose m does not divide n, then by the Division Theorem, there exists q and r integers such that:

(1) ... n=mq+r, where 0<r<m

As n is a multiple of a and b, there exists s and t integers such that:

sa=n and tb=n

Same thing happens to m as it is the least common multiple, there exists u and v such that:

ua=m and vb=m

So (1) has the following form:

n=mq+r ⇒ sa=uaq+r ⇒sa-uaq=r⇒(s-uq)a=r and

n=mq+r ⇒ tb=vbq+r ⇒ tb-vbq=r⇒ (t-vq)b=r

So r is a multiple of a and b, but r<m which is a contradiction as, m is the least common multiple of a and b. So this concludes the proof.

So this means that $\frac{ab}{m}$ is and integer.

As m= vb, then $\frac{m}{b}$ is an integer, lets say $\frac{m}{b}$ =v; and as m=ua, then $\frac{m}{a}$ =u.

So $\frac{ab}{m}$ v= $\frac{ab}{m}$ $\frac{m}{b}$ =a, so $\frac{ab}{m}$ divides a; on the other hand, $\frac{ab}{m}$ u= $\frac{ab}{m}$ $\frac{m}{a}$ =b, so $\frac{ab}{m}$ divides b. From this we can conclude that $\frac{ab}{m}$ is a common divisor of a and b.

4 0

3 years ago

This is embarrassing...help❤️❤️❤️❤️

BaLLatris [955]

Answer:

Step-by-step explanation:

P(green) is asking what are the chances that if your stuck your hand into the bag and grabbed a marble that it would be green. There are 3 green, 2 blue, 1 red, and 1 white marble for a total of 7 marbles. If you pull out a marble without looking you have 3 chances of 7 that it will be green. So

10. 3/7 or 3:7

Do we even have any yellow marbles? Not that I can see, so if there isn't a yellow to pull out, your chances of pulling one out is 0. So

11. 0/7 or 0

Since there are 3 green and 1 red, which is a total of 4 marbles, we have 4 out of 7 chances that the marble we pull is red or green. So

12. 4/7 or 4:7

7 0

3 years ago

Part B:

Andrew [12]

Articles of clothing that the store received last Thursday is 84

<u>Solution:</u>

Given that

A clothing donation store received numerous articles of clothing on Thursday of last week.

The number of articles of clothing that the store received last Thursday is represented by the following expression:

4 (3x − 1) − 2 (6 + 2x)

Need to calculate articles of clothing that the store received last Thursday when x = 12.5

As on Thursday x = 12.5, need to substitute x = 12.5 in above equation to get number of articles received on Thursday.

On substituting x = 12.5, we get

4 ( (3 x 12.5 ) – 1 ) – 2 ( 6 + ( 2 x 12.5 ) )

=> 4 (37.5 – 1) – 2 (6 + 25)

=> 146 – 62 = 84

On substituting x = 12.5 , we are getting value of expression as 84.

Hence articles of clothing that the store received last Thursday is 84

3 0

3 years ago

A rectangular parking lot has a length that is 5 yards greater than the width the area of the parking lot is 300 yd.² find the l

ArbitrLikvidat [17]

Answer:

width=5yd and length=10yd

Step-by-step explanation:

To solve this problem, you need to start by asking what you do and don't know (always where to start with word problems)

We know that Area (A) = Length (L) * Width (W)

You know Area (A) = 50 sq. yds.

You don't know anything about the width, so let's just say Width (W) = x

And you know that the length is 5 yds longer than the width, so L = W + 5yd and, since W = x, we can say L = x + 5yd

Now, we recall that A = L * W, and, since A = 50, we can say 50 sq yd = L * W.

Well, now just plug in. What does W equal and L equal?

From above we plug in and can say that 50 sq yd = x * (x + 5 yd).

Multiply it out and you get 50 = x^2 + 5x (removing units for a moment so they don't clutter the lines).

Now there are a number of ways to solve this. You can tell (because of the "x^2") that it is quadratic. So let's get it equal to zero:

0 = x^2 + 5x - 50

Now you can factor it, use your quadratic formula, or (if you hated yourself) do it as a completing the square problem. I'm going to factor.

I can see from going through a list in my head that the factors 10 and -5 multiply to -50 and add to 5, so I know that

0 = x^2 + 5x - 50 = (x + 10)(x - 5)

And so you know that x = -10 or 5.

Well, you know the parking lot cannot be -10 yd long. So we must say that W = x = 5, and, since L = W + 5, then L = 5+5 = 10. So your width is 5 yd and length is 10 yd.

<h2>mark me brainlist</h2>

6 0

2 years ago

May someone help me complete the table?

Bumek [7]

1000 grams = 1 kilogram
2000 grams = 2 kilograms
4000 grams = 4 kilograms
7000 grams = 7 kilograms
8000 grams = 8 kilograms

7 0

2 years ago

Please help as soon as possible it's due tomorrow ​

Please help as soon as possible it's due tomorrow