Difference between revisions of "2017 AMC 8 Problems/Problem 19"
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− | + | Also keep in mind that number of <math>5</math>’s in <math>98!(10,000)</math> is the same as the number of trailing zeros. Number of zeros is <math>98!</math> means we need pairs of <math>5</math>’s and <math>2</math>’s; we know there will be many more <math>2</math>’s, so we seek to find number of <math>5</math>’s in <math>98!</math> which solution tells us and that is <math>22</math> factors of <math>5</math>. <math>10,000</math> has <math>4</math> trailing zeros, so it has <math>4</math> factors of <math>5</math> and <math>22 + 4 = 26</math>. | |
− | ==Solution | + | == Video Solution == |
− | + | https://youtu.be/alj9Y8jGNz8 | |
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https://youtu.be/HISL2-N5NVg?t=817 | https://youtu.be/HISL2-N5NVg?t=817 | ||
Latest revision as of 14:37, 28 June 2021
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 . Now has factors of , so there are a total of factors of .
Solution 2
Also keep in mind that number of ’s in is the same as the number of trailing zeros. Number of zeros is means we need pairs of ’s and ’s; we know there will be many more ’s, so we seek to find number of ’s in which solution tells us and that is factors of . has trailing zeros, so it has factors of and .
Video Solution
https://youtu.be/HISL2-N5NVg?t=817
~ pi_is_3.14
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.