2017 AMC 8 Problems/Problem 19
Contents
Problem
For any positive integer , the notation denotes the product of the integers through . What is the largest integer for which is a factor of the sum ?
Solution 1
Factoring out , we have , which is . Next, has factors of . The is because of all the multiples of .The is because of all the multiples of . Now, has factors of , so there are a total of factors of .
Solution 2
Also, keep in mind that the number of ’s in is the same as the number of trailing zeros. The number of zeros is , which means we need pairs of ’s and ’s; we know there will be many more ’s, so we seek to find the number of ’s in , which the solution tells us. And, that is factors of . has trailing zeros, so it has factors of and .
Video Solution (CREATIVE THINKING + ANALYSIS!!!)
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Video Solution by OmegaLearn
https://youtu.be/HISL2-N5NVg?t=817
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Video Solution
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See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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