If the last digits (unit places) of the products $$1 \cdot 2$$, $$2 \cdot 3$$, $$3 \cdot 4$$, $$\ldots$$, $$n(n + 1)$$ are added, the result is 2010. How many products are used?

Correct Answer: 1003

Solution

The units digits of $$n(n + 1)$$ repeat every 10 terms as 2, 6, 2, 0, 0, 2, 6, 2, 0, 0, which add up to 20. After 1000 products the sum is $$100 \times 20 = 2000$$. The next three products add $$2 + 6 + 2 = 10$$, reaching 2010, so 1003 products are used.

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