B

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

An electronics company polled 300 random people to find out whether they own cell phones and laptops. The results are shown in t

kotykmax [81]

The cell labelled a is a joint frequency of 48 and the cell labelled b is a marginal frequency of 93.

<h3>What is a frequency table?</h3>

The frequency of an occurrence or a value is the number of times it happens. A frequency table is a list of objects with the frequency of each item shown in the table.

The data is mentioned in the table given below:

We need to find the value of a and b.

Here, Cell Phone+No Cell Phone=Total

To find a:

Add the no lap entries, that is 159+a=207

⇒a=48

To find b:

Add the lap entries, that is 82+11=b.

⇒b=93.

Therefore, the cell labelled a is a joint frequency of 48 and the cell labelled b is a marginal frequency of 93.

To learn more about the frequency table visit:

brainly.com/question/12576014.