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1×3×5×7×9×11×13×...×2007×2009 what is the value of the digit in the ones place of the following?

I can't find the solution for this problem. Please give me some hints

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3 Answers 3

I might be wrong but 5 by odd numbers is always 5. Last digit should be 5.

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The first thing you should look for is the $5$. Regroup all the other terms in the product:

$$ 5 \cdot (1 \cdot 3 \cdot 7 \cdot 9 \cdot 11 \cdot 13 \ldots 2009) $$

$5 \cdot 1=5$, $5 \cdot 2=0$, $5 \cdot 3=15$, $ \cdots $

You should note that if the number being multiplied by $5$ is odd, then the one's digit is 5. Likewise, if it is even, the one's digit is $0$. The product of odd numbers is always odd, so we have $5 \cdot \text{some big odd number} $, and your answer is $\color{green}{5}$.

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What does 3x5x7x9 end in? call that digit y.

What do y, yxy, yxyxy, ... ? end in.

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