Question

miki has 104 nickels and 88 dimes. she wants to divide her coins into groups where each group has the same number of nickels and the same number of dimes. what is the largest number of groups she can have?

13
11
8
4

Explanation:

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"concepts_used": [
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</pre_analysis>

<reasoning>

Identify the mathematical goal

To find the largest number of identical groups Miki can make, we need to find the greatest common divisor of her coin counts.

Find the prime factorization of 104

Using the Prime Factorization knowledge point
\[
104 = 2^3 \times 13
\]

Find the prime factorization of 88

Using the Prime Factorization knowledge point
\[
88 = 2^3 \times 11
\]

Determine the greatest common factor

Using the Greatest Common Factor knowledge point
\[
\text{GCF}(104, 88) = 2^3 = 8
\]

Verify the group distribution

We divide each coin count by the greatest common factor to ensure whole numbers.
\[

$$\begin{aligned} 104 \div 8 &= 13 \text{ nickels per group} \\ 88 \div 8 &= 11 \text{ dimes per group} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>13</mcq-option>
<mcq-option>11</mcq-option>
<mcq-correct>8</mcq-correct>
<mcq-option>4</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Greatest Common Factor"
]
}
</post_analysis>

Answer:

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"step_cot",
"multiple_choice_formatter",
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"concepts_used": [
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"Prime Factorization"
],
"new_concepts": [],
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}
</pre_analysis>

<reasoning>

Identify the mathematical goal

To find the largest number of identical groups Miki can make, we need to find the greatest common divisor of her coin counts.

Find the prime factorization of 104

Using the Prime Factorization knowledge point
\[
104 = 2^3 \times 13
\]

Find the prime factorization of 88

Using the Prime Factorization knowledge point
\[
88 = 2^3 \times 11
\]

Determine the greatest common factor

Using the Greatest Common Factor knowledge point
\[
\text{GCF}(104, 88) = 2^3 = 8
\]

Verify the group distribution

We divide each coin count by the greatest common factor to ensure whole numbers.
\[

$$\begin{aligned} 104 \div 8 &= 13 \text{ nickels per group} \\ 88 \div 8 &= 11 \text{ dimes per group} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>13</mcq-option>
<mcq-option>11</mcq-option>
<mcq-correct>8</mcq-correct>
<mcq-option>4</mcq-option>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Greatest Common Factor"
]
}
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