I dont Understand this so can anyone help me please!! Thank you

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

The corners of a square lie on a circle of diameter d = 0.35 m. each side of the square has a length l. find l.

sergeinik [125]

Refer to the attached image.

As we can observe ABCD is a square, whose corners that is A,B,C and D lie on the circle.

The diameter AC = 0.35 meter.

We have to find the side of square.

Let the measure of side of a square be 'l' meter

In triangle ABC,

By Pythagoras theorem, we get

$(AC)^2 = (AB)^2 + (BC)^2$

$(0.35)^2 = (l)^2 + (l)^2$

$(0.35)^2 = 2(l)^2$

$\frac{(0.35)^2}{2} = (l)^2$

$l^2 = 0.06125$

So, l = 0.25 meter

Therefore, the measure of side of the square is 0.25 meter.

6 0

3 years ago

The sales price of a single family house in Charlotte is normally distributed with mean $210,000 and standard deviation $35,000.

Lady_Fox [76]

Answer:

E(X ¯)=210,000.

Step-by-step explanation:

A sampling distribution for samples of size n=49 from a population with means μ=210,000 and standard deviation σ=35,000, has the following means anda standard deviation:

$\mu_s=\mu=210,000\\\\\sigma_s=\sigma/\sqrt{n}=35,000/\sqrt{49}=35,000/7=5,000$