NotThatSmart

Let's write a system of equations to solve this problem. 

Let's let G be Grogg's age and D be the dad's CURRENt age. 

From the first part of the problem, we ge tthe equation

\(9(g-7)=d-7\\ 9(g-7)+7=d\)

From the second part of the problem, we have the equation 

\(6(g-4)=d-4\\ 6(g-4)+4=d\)

Now, since both equations are equal to d, we can set the right side to equal each other. 

Thus, we have the equation

\(9(g-7)+7=6(g-4)+4\)

Now, we can find Grogg's age. We find that Grogg is \(g=12\)

Plugging this value back into the first equation, we get

\(d=9(12-7)+7=9(5)+7=52\)

So Grogg's dad is 52 years old. 

Thanks! :)

MY 600th ANSWER

The answer is \(A, B, C = (1, 1, 1)\)

We get the monstrosity \(10101001010100010111000110010101100011_2\), which when converted to base 10, we get

\(181803967843_{10}\)

This divided by 70 (it would go to scientific notaiton if not) gets us \(229488030591\), 

meaning it is divisible by 7. 

Thanks! :)

Let's create a system of equations to solve this problem .

Let's let A be the number of money Alice has and B be the amount of money Bob has. We have

If Alice receives n dollars from Bob, then she will have 3 times as much money as Bob.

\(A + n  =  3 * (B - n) \)

If she gives n dollars to Bob, then she will have 2 times as much money as Bob.

\(A - n  =  2 * (B + n) \)

Thus, we have the system

\(A + n = 3B - 3n → A = 3B - 4n → 3A = 9B - 12n \\ A - n = 2B + 2n → A = 3B + 3n → 4A = 12B + 12n\)

adding these two equations up, we get

\(7A = 17B → 7A / B = 17 → A / B = 17 / 7 \)

So, the answer is \(17/7\)

Thanks! :)

We can write a handy equation to solve this problem. 

First, let's set a variable. Let's set x as a positive integer. 

Any number congruent to 7 modulo 29 can be written in the form

\(29x+7\) since it is 7 greater than a factor of 29. 

thus, we can write the inequality

\(29x+7\leq 1000\)

Setting the two to equal and solving for x, we have

\(29x+7=1000\\ 29x=993\\ x \approx 34.24\)

Rounding down, we find x is 34. 

So our answer is 34. 

Thanks! :)

the answer is \(34\)

The answer is \(244224 \pmod {64} \equiv 0 \)

\((n-3) + (n-2) + (n-1) + n + (n+1) + (n+2) + (n+ 3) = (28/9)(n + 3) \\ 7n = (28/9)(n + 3)\\ 7n = (28/9)n + 28/3 \\ 63n = 28n + 84 \\ 63n - 28n = 84 \\ 35n = 84\\ n = 84 / 35 = 12 / 5 \\ (n - 3) = (12/5 - 3) = -3 / 5\)

This isn't an integer, but

the answer is \( -3 / 5 \)

Thanks! :)

For N to satisfy all the choices given, we must find the LCM of 6, 7, and 8. 

We have that \(LCM[6,7,8]=168\)

Since they are all containing remainder 2, we have the equation

\(168x+2 \leq 1000\)

Since we want the max number, we set the equations to equal each other and solve for x. We get

\(168x+2=1000\\ 168x=998\\ x=\frac{499}{84} \approx 5.940\)

We round down, so our final answer is 5. 

Thanks! :)

First, let's use the FOIL method and combine two of the terms together. 

Combining two at a time, we get

\(n\left(n^2+16n-17\right)\left(n^2-54n+693\right)\)

Then, once again, combining the trinomials, we get

\(n\left(n^4-38n^3-188n^2+12006n-11781\right)\)

Distributing in the n. we finally find that

The final answwer is \(n^{5}-38n^{4}-188n^{3}+12006n^{2}-11781n\)

Thanksd! :)

Let's work from bottom up. Starting with the smallest, we have \(\frac{1}{\frac{17}{4}}=\frac{4}{17}\)

Then, we have \(4+\frac{4}{17}=\frac{72}{17}\)

\(4+\frac{1}{4+\frac{1}{4}}=\frac{72}{17}\)

Thus, we have

\(\frac{1}{4+\frac{1}{4+\frac{1}{4}}}=4+\frac{17}{72}=\frac{305}{72}\)

The final answer is \(\frac{305}{72}\)

Let's focus on the two smallest numbers first. 15 and 24. 

Since N leaves a remainder, we can elmiinate the factors. There are

Thus, N cannot be \( 3, 5, 2, 4, 6, 8, 12\)

This leaves 7, 9, 10, 11, 13, and 14.   Now, let's test out all the other numbers

15/7 = 2 R 1      24/7 = 3 R 3      so it isn't 7     

15/9 = 1 R 6      24/9 = 2 R 6      so 9 looks hopeful, we'll try another   

1457/9 = 161 R 8                        so 9 is eliminated   

15/10 = 1 R 5     24/10 = 2 R 4    so it isn't 10   

15/11 = 1 R 4     24/11 = 2 R 2     so it isn't 11   

15/13 = 1 R 2     24/13 = 1 R 11   so it isn't 13   

15/14 = 1 R 1     24/14 = 1 R 10   so it isn't 14   

mmm...I don't think there is a solution

Thanks! :)