We see:

\(\sqrt{100}=10;\sqrt{999}>31\)

So the numbers from 10 to 31 is the square root of three-digit positive intergers 

So there are \(31-10+1=22\) numbers

We have:

\(\left[{}\begin{matrix}\overline{6D45}⋮3\\0\le D\le9\end{matrix}\right.\)

But 6 + 4 + 5 = 15 divisived by 3

So D must divisived by 3 and \(0\le D\le9\)

=> \(D\in\left\{0;3;6;9\right\}\)

So their sum is:

0 + 3 + 6 + 9 = 18

\(\dfrac{1}{m}+\dfrac{1}{2m}=\dfrac{2m}{2m^2}+\dfrac{m}{2m^2}=\dfrac{3m}{2m^2}=\dfrac{3}{2m}=6\)

\(\Rightarrow\dfrac{3}{2m}=6\Rightarrow2m=\dfrac{1}{2}\Rightarrow m=\dfrac{1}{4}\)

\(K=1+3+7+...+107\)

\(=\left(2^1-1\right)+\left(2^2-1\right)+\left(2^3-1\right)+...+\left(2^6-1\right)+107\)

\(=1+3+7+15+31+63+107=227\)

1.

Because p and q are prime number > 3

=> q can equal 3k+1 or 3k+2

But if q = 3k+1 => p = 3k+1 + 2 = 3k+3 divided by 3 -> wrong

And if q = 3k+2 => p = 3k+2 + 2 = 3k+4 divided by 3 remainder 1   -> right

So p+q = 3k+2 + 3k+4 = 6k+6 = \(6\left(k+1\right)⋮6\)

We have 2 cases:

\(\left[{}\begin{matrix}k=2h+1\\k=2k\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}6\left(k+1\right)=6\cdot\left(2h+2\right)=12h+12⋮12\\6\left(k+1\right)=6\left(2h+1\right)=12h+6\equiv6\left(mod12\right)\end{matrix}\right.\)So p+q divided by 12 remainder 0 or 6

\(\dfrac{a\dfrac{1}{b}}{b\dfrac{1}{a}}=\dfrac{a+\dfrac{1}{b}}{b+\dfrac{1}{a}}=\dfrac{\dfrac{ab}{b}+\dfrac{1}{b}}{\dfrac{ab}{a}+\dfrac{1}{a}}=\dfrac{\dfrac{ab+1}{b}}{\dfrac{ab+1}{a}}=\dfrac{ab+1}{b}\cdot\dfrac{a}{ab+1}=\dfrac{a}{b}\)

So \(\dfrac{a\dfrac{1}{b}}{b\dfrac{1}{a}}=\dfrac{a}{b}\)

Easy ^.^

7/4 = 1.75

So their positive difference between 3.75 and 7/4 is:

3.75 - 1.75 = 2

The degree measure of supplement of its third angle:

\(180-\left(180-7-97\right)=104\) [degree]

If Phara buy all items, she must cost:

$2.99 + $3.49 + $6.29 + $4.99 + $3.89 + $5.49 = $27.14

So she it took more: $27.14 - $17.36 = $9.78

But $9.78 is the sum money must cost of Puzzle and Wallet.

So she didn't bought Puzzle and Wallet

So the positive difference of their is:

$6.29 - $3.49 = $2.8

The smallest prime number has 2 digits is 11

The greatest prime number has 2 digits is 97

So their product is:

11.97 = 1067

The half perimeter of the garden is:

48: 2 = 24 [ft]

The length of this garden is:

\(24:\left(1+3\right)\cdot3=18\left(ft\right)\)

The width of this garden is:

24 - 18 = 6 [ft]

So the measure of the diagonal of the garden is:

\(\sqrt{18^2+6^2}=6\sqrt{10}=1897,366596\%\)

To complete a dozen cookies required:

4/5 = 0,8 [cups of flour]

So to complete 15 dozen cookies she need:

0,8 . 15 = 12 [cups of flour]

Answer: 12 cups

There are \(\dfrac{1421}{7}\cdot4=812\) dentists recommended Minty Fresh Mouthwash

Let a,b,c sequence is length, width and height of this rectangular prism. We have:

\(\dfrac{a}{3}=\dfrac{b}{2}=c\); abc = 48

Let \(\dfrac{a}{3}=\dfrac{b}{2}=c=k\Rightarrow\left\{{}\begin{matrix}a=3k\\b=2k\\c=k\end{matrix}\right.\)

=> abc = 3k.2k.k = 6k³ = 48

=> k³ = 8 => k = 2

=> a = 2.3 = 6 

So the length of this prism is 6in = 15,24cm 

The perimeter of the hexagon is:

6.6 = 36 [cm]

So perimeter of this hexagon is 36cm

The time now is: 

10:30 + 0:55 = 11:25

So the time now to noon is:

12:00 - 11:25 = 0:35

Answer: 35 minutes

The number of % of the grapes are not rad are:

100% - 20% = 80%

So the number of the grapes are not red are:

50.80% = 40 [grapes]

So there are 40 grapes are not red

Done ^.^

The smallest odd between 20 and 158 is 21

The greatest odd between 20 and 158 is 157

So the number of odd between 20 and 158 is:

\(\dfrac{157-21}{2}+1=69\left(odd-numbers\right)\)

So there are 69 odd number between 20 and 158

The yellow tint of first mixture is:

30.30% = 9 [litters]

The yellow tint of after mixture is:

9 + 5 = 14 [liters]

The liters of after mixture is:

30 + 5 = 35 [liters]

The percent of yellow tint in the new mixture is:

14: 35.100 = 40%

So choose C

This number is:

12 : 20% = 60

So 30% of this number is:

60 . 30% = 18

So choose B