Bot GPT 4-o mini Let's denote the number of packs of hotdogs John bought as \( h \) and the number of packs of buns as \( b \). According to the problem, each pack of hotdogs contains 9 hotdogs, and each pack of buns contains 7 buns. Since John has the same number of hotdogs as buns, we can express this relationship mathematically:
\[ 9h = 7b \]
To find the least number of hotdogs and buns, we need the smallest positive integers \( h \) and \( b \) for which this equation holds. Rearranging the equation gives us:
\[ \frac{h}{b} = \frac{7}{9} \]
This indicates that \( h \) and \( b \) can be expressed in terms of a common multiple. We can let:
\[ h = 7k \quad \text{and} \quad b = 9k \]
for some integer \( k \). Substituting back into the expression for the number of hotdogs and buns gives:
\[ \text{Number of hotdogs} = 9h = 9(7k) = 63k \]
\[ \text{Number of buns} = 7b = 7(9k) = 63k \]
Both approaches yield the same total number, \( 63k \), meaning the least common case occurs when \( k = 1 \).
Thus, the least number of hotdogs and buns is 63, and they can be calculated as follows:
- Number of hotdogs: \( 63 \)
- Number of buns: \( 63 \)
Now calculating the number of packs:
- Number of packs of hotdogs \( h = 7 \) (since \( 63 \div 9 = 7 \))
- Number of packs of buns \( b = 9 \) (since \( 63 \div 7 = 9 \))
In summary, John bought:
- 63 hotdogs
- 7 packs of hotdogs
- 9 packs of buns
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