All Answers 358090 Answers

$$\small{\text{
Find the sum of all of the integers
}}\\
\small{\text{
in the set $\{-18,27,54,254,754,1036,3254\}$
}}\\
\small{\text{
that are congruent to 6 modulo 8.
}}$$

$$\small{\text{$
\begin{array}{rcl}
-18 \equiv -2 & \equiv & 6 \pmod{8} \\
27 \equiv 3 & \not\equiv & 6 \pmod{8} \\
54 \equiv 6 & \equiv & 6 \pmod{8} \\
254 \equiv 6 & \equiv & 6 \pmod{8} \\
754 \equiv 2 & \not\equiv & 6 \pmod{8} \\
1036 \equiv 4 & \not\equiv & 6 \pmod{8} \\
3254 \equiv 6 & \equiv & 6 \pmod{8} \\
\end{array}
$}}$$

the sum is -18 + 54 + 254 + 3254 = 3544

Thanks Geno, I thought strain would have some kind of  unit attached to it  

Is that a normal way to do it?

Ok, it is not a problem sabi,  

I am sure you will get yourself better organised.   

$$\small{\text{
Find the units digit of $3^{777}$ or $3^{777} = x \pmod {10} $
}}$$

$$\small{\text{$
\begin{array}{lrcll}
1. & ~~gcd(10,3) &=& 1 \qquad & | \qquad gcd = \mathrm{greatest~common~ divisor}\\
2. & ~~\varphi{(10)} &=& 4 \qquad & | \qquad \varphi = \mathrm{Euler ~function}\qquad \varphi{(10)} = 10 \cdot (1-\dfrac{1}{2})\cdot (1-\dfrac{1}{5}) \\
3. & ~~a^{\varphi (n)} &\equiv& 1 \pmod{n} \qquad & | \qquad \mathrm{Euler's ~Theorem,~~ if~ } gcd(a,n)=1 \qquad \\
& ~~ 3^4 &\equiv& 1 \pmod {10} \qquad & | \qquad n = 10\qquad a = 3\\\\
& & 3^{777} \pmod{10}\\\\
& &\equiv& 3^{4\cdot 194 +1} \pmod{10}\\
& &\equiv& 3^{4\cdot 194}\cdot 3 \pmod{10}\\
& &\equiv& (3^4)^{194}\cdot 3 \pmod{10}\\
& &\equiv& (\underbrace{3^4}_{=1 \pmod{10}})^{194}\cdot 3 \pmod{10}\\\\
& &\equiv& 1^{194}\cdot 3 \pmod{10}\\
& &\equiv & \textcolor[rgb]{1,0,0}{3} \pmod{10}\\ \\
& \mathbf{the~ Units~ digit~ of ~} 3^{777} \mathbf{ ~is ~} 3\\
\end{array}
$ }}$$

I think the expected number of coupons will be 0.6n, where n is the number of visits of the regional manager.

.

In that case feel free to use 0/0, but don't expect to pass too many mathematics exams!

.

okay! but 1 can equal 2

1 banana = 2 ends

so it could be right

I'm not sure if you really want to know or if you've set it as a puzzle! Either way:

.

"Undefined" means there is no value for 0/0 in mathematics.  Expressions that contain or result in 0/0 are not mathematically valid.   The reason is that it can lead to illogical results (you could use it to show that 1 equals 2 for example!).

.

What is the definition of undefined, thanks for the answer.

This simplifies to 12n² + 5

To evaluate this you need to know the value of n

.

This is undefined.

.

When looking for the units digit of 3ⁿ, look at the value of mod(n,4).  When this is 0, the units digit is 1, when it is 1 the units digit is 3, when it is 2 the units digit is 9, when it is 3 the units digit is 7.

mod(777,4) = 1, so the units digit of 3⁷⁷⁷ is 3.

.

Chill yourself Titanium, there is no fight happening here, especially not by me :)

$$\small{\text{
If $n \equiv 43 \pmod{60}$, then what is the residue of $n$ modulo 6?
}}$$

$$\small{\text{$
\begin{array}{lrcl}
(1) & n - 43 &=& 60\\
(2) & n-x &=& m\cdot 6 \\
\\
\hline
\\
(1)-(2) & n-43 - n + x &=& 60 - m\cdot 6 \\
& -43 + x &=& 60 - m\cdot 6 \\
&x &=& 60+43- m\cdot 6 \\
&x &=& 103- m\cdot 6 \qquad | \qquad m =17 \mathrm{~~lowest
~positive ~integer}\\
&x &=& 103-102\\
& \mathbf{x} & \mathbf{=} & \mathbf{1 }
\end{array}
$}}\\$$

the residue of n modulo 6 is 1

0.6 I guess.    

I mean he can only get 0 or 1 if he only serves the manager once.  But t he expected return in a statistical sence is 0.6 I think. 

- An airplane needs to travel 400 km. Determine a function for the speed of the airplane, with respect to time.

f(t)=400/t           where t=time    and f(t)=speed   (km/time unit)

-Scott wants to calculate the distance from his house to each of his friends' houses. If he drives at 50 km/h, find a function for the distance, with respect to the number of hours it takes to travel.

f(h)=50h       where h=hours and   f(h) is distance in km

Jenny has 8 stamp books that each contain 42 pages. Each page in her books contains 6 stamps. Jenny decides to reorganize her stamp books such that each page contains 10 stamps. This will give her more space to collect stamps without having to buy new books. Under her new system, Jenny fills up 4 complete books, still with 42 pages per book. Her fifth book now contains 33 pages filled with 10 stamps per page and 1 last page with the remaining stamps. How many stamps are on that last page ?

$$\small{\text{$
\begin{array}{lrcl}
\\
\mathrm{all~ stamps}: & 8~ \mathrm{books} \cdot 42 ~\mathrm{pages}\cdot 6 ~\mathrm{stamps}&=& 2016 ~\mathrm{stamps.} \\\\
\mathrm{now}: & x~ \mathrm{books} \cdot 42 ~\mathrm{pages}\cdot 10 ~\mathrm{stamps}&=& 2016 ~\mathrm{stamps.} \\\\
& x &=& \dfrac{2016}{420}\\\\
&\mathbf{x} & \mathbf{=} & \mathbf{4.8 ~\mathrm{books}}\\
\end{array}
$}}\\$$

Jenny has 4 complete books. Remaining a 0.8 book    -     0.8 * 42 = 33.6 pages

Jenny has 33 complete pages. Remaining a 0.6 page   -    0.6 * 10 = 6 stamps

Jenny has 6 stamps on that last page

Thanks Heureka,

I am going to do it a little differently but our answers are the same :)

A 9x12 rectangle is inscribed in a circle. What is the radius of the circle?

The diagonal of the rectangle is the hypotenuse of the triangle with short sides 9 and 12

It is also the diameter of the circle.

$$\\d^2=9^2+12^2\\
d^2=81+144\\
d^2=225\\
d=15\\
$The radius is half the diameter so radius = 7.5 units$$$

You might have got this straight of if you had realised that

3*3=9

4*3=12

5*3=15

3,4,5 is a pythagorean triad so, so is 9,12,15.     So the diameter had to be 15units

A 9x12 rectangle is inscribed in a circle. What is the radius of the circle ?

$$\small{\text{$
\begin{array}{rcl}
r^2 &=& \left(\dfrac{9}{2}\right)^2+ \left(\dfrac{12}{2}\right)^2\\\\
r^2 &=& 4.5^2+6^2\\\\
r^2 &=& 56.25 \\\\
r &=& \sqrt{56.25 }\\\\
\mathbf{r} & \mathbf{=} & \mathbf{7.5}
\end{array}
$}}$$

Yes, I saw it but it did not look very helpful to the question asker.

Maybe these might be more useful

T=-8+2h

Where T is the temp and H is the number of hours that have passed since the original early reading of -8degrees.

Obviously there needs to be constraints,  h must be positive, and h obviously needs an upper constraint as well, the temp will not keep going up for more than a few hours, but anyway that is probably the simplified answer that you need.

$$\small{\text{
What integer $n$ satisfies $0\le n<9$ and $-1111\equiv n\pmod 9~?$
}}$$

$$\small{\text{
$-1111 \pmod 9 \equiv -4 \equiv -4+9 \equiv 5$
}}$$

n= 5