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 .
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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 .
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Solution 3
We can first factor a out of the to get Simplify to get .
Let's first find how many factors of has. is because is . After we remove the brackets, we get , and . We only care about the latter (second one), because the problem only ask's for the power of . We get
Next, we can look at the multiples of 5 in . so there is 19 multiples of 5. We get
But we cannot forget the multiples of with fives in it. Multiples of . How many multiples of are between and ? . and that's it. We get
Finally, we add all of the numbers (powers of ) up. That is , which is just
So the answer is . Which is answer choice D .
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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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