To determine the greatest number of pencil sets the store can sell, we need to divide the total number of pencils (634) by the number of pencils in each set (15).
We perform the division:
\[ \text{Number of pencil sets} = \frac{634}{15} \]
Calculating this gives:
\[ 634 \div 15 = 42.2667 \]
Since the store cannot sell a fraction of a pencil set, we take the greatest integer less than or equal to \(42.2667\), which is 42.
To confirm, we can calculate how many pencils are used in 42 sets:
\[ 42 \times 15 = 630 \]
This means that 630 pencils are used, leaving:
\[ 634 - 630 = 4 \text{ pencils} \]
Thus, the store can sell a maximum of 42 pencil sets.
Therefore, the greatest number of pencil sets the store can sell is \( \boxed{42} \).
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