RylanSchaeffer
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1 month ago Open RylanSchaeffer opened 1 month ago
RylanSchaeffer
commented
1 month ago RylanSchaeffer
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1 month ago The file eval_results/math_samples/Pythia_1B_300B/152.yaml:
gt_answer: For all $n$, $1 + 2 + \dots + n = n(n + 1)/2$, so $1+2+3+\dots+80=\frac{80
\cdot 81}{2}=40\cdot81=2^3\cdot5\cdot3^4$. Thus, the greatest prime factor of the
sum is $\boxed{5}$.
prompt: 'Problem:
Find the domain of the expression $\frac{\sqrt{x-2}}{\sqrt{5-x}}$.}
Solution:
The expressions inside each square root must be non-negative. Therefore, $x-2 \ge
0$, so $x\ge2$, and $5 - x \ge 0$, so $x \le 5$. Also, the denominator cannot be
equal to zero, so $5-x>0$, which gives $x<5$. Therefore, the domain of the expression
is $\boxed{[2,5)}$.
Final Answer: The final answer is $[2,5)$. I hope it is correct.
Problem:
If $\det \mathbf{A} = 2$ and $\det \mathbf{B} = 12,$ then find $\det (\mathbf{A}
\mathbf{B}).$
Solution:
We have that $\det (\mathbf{A} \mathbf{B}) = (\det \mathbf{A})(\det \mathbf{B})
= (2)(12) = \boxed{24}.$
Final Answer: The final answer is $24$. I hope it is correct.
Problem:
Terrell usually lifts two 20-pound weights 12 times. If he uses two 15-pound weights
instead, how many times must Terrell lift them in order to lift the same total weight?
Solution:
If Terrell lifts two 20-pound weights 12 times, he lifts a total of $2\cdot 12\cdot20=480$
pounds of weight. If he lifts two 15-pound weights instead for $n$ times, he will
lift a total of $2\cdot15\cdot n=30n$ pounds of weight. Equating this to 480 pounds,
we can solve for $n$:
\begin{align*}
30n&=480\
\Rightarrow\qquad n&=480/30=\boxed{16}
\end{align*}
Final Answer: The final answer is $16$. I hope it is correct.
Problem:
If the system of equations
\begin{align*}
6x-4y&=a,\
6y-9x &=b.
\end{align*}has a solution $(x, y)$ where $x$ and $y$ are both nonzero,
find $\frac{a}{b},$ assuming $b$ is nonzero.
Solution:
If we multiply the first equation by $-\frac{3}{2}$, we obtain
$$6y-9x=-\frac{3}{2}a.$$Since we also know that $6y-9x=b$, we have
$$-\frac{3}{2}a=b\Rightarrow\frac{a}{b}=\boxed{-\frac{2}{3}}.$$
Final Answer: The final answer is $-\frac{2}{3}$. I hope it is correct.
Problem:
What is the greatest prime factor of the sum of the arithmetic sequence $1 + 2 +
3 + \cdots + 80$?
Solution:'
question: What is the greatest prime factor of the sum of the arithmetic sequence
$1 + 2 + 3 + \cdots + 80$?
samples:
- '
We know that $1 + 2 + 3 + \cdots + 80 = \frac{1}{2}(1 + 2 + 3 + \cdots + 80) = \boxed{2^{80}}$.
Final Answer: The final answer is $2^{80}$. I hope it is correct.
'
- '
The sum of the sequence $1 + 2 + 3 + \cdots + 80$ is
$$\sum_{n=1}^{80}n=\boxed{8192}.$$
The greatest prime factor of $8192$ is $2$.
'
$p=3$, we have $p=5$, so $p=5$. If $p=5$,
we have $p=7$, so $p=7$. Therefore, the greatest prime factor of $p-1$ is $p-1=2$. Thus,
$p=2$ is the greatest prime factor of the sum. Therefore, the greatest prime factor
of the sum is $p=2$.
Final Answer: The final answer is $2$. I hope it is correct.
'