NotThatSmart

The answer is $3$

$40 \cdot 6 \cdot 75 \cdot 12=216000$

This number has 3 terminal zeroes. 

Thanks! :)

First, let's make a few observations. 

A power of 5 always ends with a 5. 

$7^{13}=96889010407$, so 7^13 ends on a 7, 

And 23 ends with a 3. 

So we have $7+5+3 = 15$. This means that $5^{19} + 7^{13} + 23$ is divisible by 5.

Since 5, 7, and 23 are all odd, it's not possible the sum is divisble by 2. 

Also, since none of the terms are divisible by 3, the sum is not divisble by 3. 

Thus, 5 is our answer. 

Thanks! :)

Let's set up for a really cool function. 

First, note that

$1200 = 12 * 5^2 * 2^2 = 2^4 * 3 * 5^2$

Now, we use Euler's Totient Function. We have

$\phi{1200} = 1200 / ( 2 * 3 * 5) * (2-1)(3-1) (5-1) = 40 (1)(2)(4) = 320$

The answer is $320$

Thanks! :)

Let's set up for a really cool function. 

First, note that

$1200 = 12 * 5^2 * 2^2 = 2^4 * 3 * 5^2$

Now, we use Euler's Totient Function. We have

$\phi{1200} = 1200 / ( 2 * 3 * 5) * (2-1)(3-1) (5-1) = 40 (1)(2)(4) = 320$

The answer is $320$

Thanks! :)

The answer is $4$

1*1 = 1

2*2 = 4

Thanks! :)

Let the first machine cycle two times and then shut it off, for a total of 10 added.   

Let the second machine cycle three times and then shut it off, for a total of 9 removed.   

10 – 9 = 1 candy left in the bottom    

Thus, the answer is $2$

Thanks! :)

Let's look at the number 512. 

We know that $512 = 2^9$, which means it has 10 divisors. 
Since it's divisble by 2 and two is the smallest prime, 

our final answer is 2. 

Thanks! :)

Let's first take a look at the fraction 1/17. 

It has 16 repeating digits in the form of $0. \overline{0588235294117647}$

We simplfy have to find which digit the 4000th lies on. 

First, let's do 4000/16. 

We get $4000/16=250$

So the 4000th digit is the last digit of the repeating decimal, which is 7. 

So 7 is our answer. 

Thanks! :)

First, let's simplify the equation. We have

$x^2 - 6x + 13$

Since the constant is equal to uv, we have the equation

$uv = 13 \\ 2uv = 26$

The coefficient of x is equal to u+v, so we have

$u + v = 6$

Squaring both sides of the equation, we have the equation

$u^2 + 2uv + v^2 = 36 \\ u^2 + 26 + v^2 = 36 \\ u^2 + v^2 = 10$

Now, it's time for the fun part of this problem. We can simplify what we want to figure out to

We already know all these terms! We can easily find them! We have

$u^3 + v^3 = ( u + v) ( u^2 + v^2 - uv) = (6)(10 - 13) = -18 \\ [uv]^2 = [ 13]^2 = 169$

So, The answer is $u/v^2 + v/u^2 = [ u^3 + v^3 ] / [ uv]^2 = -18 / 169$

Thanks! :)

Simplify as

$10x^2 + 10x - 1\\ ab = -1/10\\ 2ab = -2/10  = -1/5$

a + b  =  -1      square both sides

a^2 + 2ab + b^2  = 1

a^2 - 1/5 + b^2  = 1

a^2 + b^2  = 1 + 1/5  =  6/5

$(a -4) / ( b -4)  +  ( b -4) / (a -4)  =     [  (a - 4)^2  + ( b -4)^2] ]  ' [ (a -4) ( b-4) ]  = \\ [ a^2 - 8a +16  + b^2 - 8b + 16 ]  /  [ ab - 4(a + b) +16 ]   = \\ [  (a^2 + b^2) - 8(a + b) + 32 ]  / [ ab - 4 (a + b) + 16 ]  =\\ [ 6/5 - 8 (-1) + 32 ]  [ -1/10 - 4 (-1) + 16 ]  =  \\ [ 6/5 + 40 ]  / [ 20 - 1/10] =   20497 / 25$

The answer is $20497 / 25$

Thanks! :)