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For the purpose of this question, a square is considered a kind of rectangle. Given the rectangle with vertices $$(0, 0)$$, $$(0, 223)$$, $$(9, 223)$$, $$(9, 0)$$, divided into 2007 unit squares by horizontal and vertical lines. By cutting off a rectangle from the given rectangle, we mean making cuts along horizontal and (or) vertical lines to produce a smaller rectangle. Let $$m$$ be the smallest positive integer such that a rectangle of area $$m$$ cannot be cut off from the given rectangle. Then $$m =$$ _________
Correct Answer: 227
A rectangle of area $$m$$ can be cut off exactly when $$m = pq$$ with $$p \le 9$$ and $$q \le 223$$. Every $$m \le 223$$ works as $$1 \times m$$, and $$224 = 2 \times 112$$, $$225 = 9 \times 25$$, $$226 = 2 \times 113$$ also work. But 227 is prime and larger than 223, so it has no such factorisation, and $$m = 227$$.
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