Write a polynomial in standard form with the zeros

All digits in a dropdown number are different, and one of its digits is the average of all its digits. It has at least two digit

Svetradugi [14.3K]

1. Start the search among 2-digit numbers. A dropdown number (DDN) with 2 digits is a number $ab$ such that

$\dfrac{a+b}2 = a \implies a + b = 2a \implies b = a$

or

$\dfrac{a+b}2 = b \implies a+b = 2b \implies a=b$

but both of these solutions violate the requirement that the digits are distinct, so there are no 2-digit DDNs.

A 3-digit DDN $abc$ is such that

$\dfrac{a+b+c}3 = a \implies a+b+c = 3a \implies b+c = 2a$

or $a+c=2b$ if the average is $b$ , or $a+b=2c$ if the average is $c$ . The smallest possible value for $a$ is 1 since we require 3 digits. Then $b+c=2$ , and we can pick $b=0$ and $c=2$ to get the smallest DDN, 102.

2. In a 4-digit DDN $abcd$ , we have

$\dfrac{a+b+c+d}4 = a \implies a + b + c + d = 4a \implies b+c+d=3a$

or $a+c+d=3b$ or $a+b+d=3c$ or $a+b+c=3d$ .

We're free to fix $a=1$ and $b=0$ to try to get the smallest DDN. This leaves us with $c+d=3$ or $c+d=-1$ or $1+d=3c$ or $c=3d$ .

The first two cases are impossible - the only choices for $c,d$ such that $c+d=3$ are 1 and 2, and the sum of two positive integers must be positive. The smallest possible value of $c$ is 2; this leaves us with $1+d=6$ or $2=3d$ , but the latter case is impossible because 3 does not divide 2. So $d=5$ , and the smallest 4-digit DDN is 1025.

To find the largest DDN, start with the largest possible values for $a$ and $b$ . Let $a=9$ and $b=8$ . Then $c+d=19$ or $c+d=15$ or $17+d=3c$ or $17+c=3d$ . At most, we can have $c+d=13$ with 7 and 6, so the first two cases are impossible. If we maximize $c=7$ , then either $17+d=21\implies d=4$ or $24=3d\implies d=8$ (which we don't want). So the largest 3-digit DDN is 9874.

3. I don't have an analytical solution to this, but using brute force (program) the total count is 112.

4. It is possible; consider 1249 and 1250, with digital averages

$\dfrac{1+2+4+9}4=4 \text{ and } \dfrac{1+2+5+0}4=2$

which happens to be the smallest pair. (Also found with brute force.)

3 0

2 years ago

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How can you determine whether two circles are congruent?

kvasek [131]

Answer:

If the diameter or radius of one circle is the same as another circle they are congruent.

7 0

3 years ago

The original line has the equation of y=-2x+8

enot [183]

The new equation of line that is perpendicular to the original that goes through the point (6, -1) in slope intercept form is $y = \frac{1}{2}x - 4$

<h3>Solution:</h3>

Given that original line has the equation of y = -2x + 8

We have to write a new equation that is perpendicular to the original that goes through the point (6, -1)

Let us first find slope of original line

The slope intercept form of line is given as:

y = mx + c ------ eqn 1

Where "m" is the slope of line and "c" is the y - intercept

On comparing the slope intercept form and given original equation, we get "m = -2"

Thus slope of original line "m" = -2

We know that product of slope of a line and slope of line perpendicular to it are always -1

slope of original line x slope of line perpendicular to it = -1

$\begin{array}{l}{-2 \times \text { slope of line perpendicular to it }=-1} \\\\ {\text { slope of line perpendicular to it }=\frac{1}{2}}\end{array}$

Let us find equation of line with slope m = 1/2 and passes through point (6, - 1)

Substitute $m = \frac{1}{2}$ and (x, y) = (6, -1) in eqn 1

$-1 = \frac{1}{2} \times 6 + c\\\\-1 = 3 + c\\\\c = -4$

Thus the required equation of line is:

Substitute "c" = -4 and $m = \frac{1}{2}$ in eqn 1

$y = \frac{1}{2}x - 4$

Thus the equation of line perpendicular to original line is found

7 0

3 years ago

A claw game holds pink, red, green, and blue stuffed toys. The machine randomly picks up a toy but then drops it back into the c

mote1985 [20]

most red toys and least green toys

4 0

3 years ago

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Christine must buy at least 45 fluid ounces of milk at the store. The store only sells milk in 200 milliliter bottles. If there

ratelena [41]

Since there are 1000 milliliters in a liter, that means that there are 33.8/1000 fluid ouncers per milliliter. That means that each 200 mil bottle is 6.76 fluid ounces or 33.8/1000*200. 45/6.76 = 6.66 but you have to round up to a whole number of bottles since you can buy 2/3 of a bottle which makes your answer 7 bottles.

8 0

3 years ago

Write a polynomial in standard form with the zeros –4, 0, 1, and 4.