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− | ==Problem==
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− | For any positive integer <math>M</math>, the notation <math>M!</math> denotes the product of the integers <math>1</math> through <math>M</math>. What is the largest integer <math>n</math> for which <math>5^n</math> is a factor of the sum <math>98!+99!+100!</math> ?
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− | <math>\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27</math>
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| ==Solution 1== | | ==Solution 1== |
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| ~CHECKMATE2021 | | ~CHECKMATE2021 |
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− | ==Solution 2==
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− | Also, keep in mind that the number of <math>5</math>’s in <math>98! (10,000)</math> is the same as the number of trailing zeros. The number of zeros is <math>98!</math>, which 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 the number of <math>5</math>’s in <math>98!</math>, which the 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 = \boxed{26}</math>.
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− | ~CHECKMATE2021
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− | ==Solution 3==
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− | We can first factor a <math>98!</math> out of the $
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| ==Video Solution (CREATIVE THINKING + ANALYSIS!!!)== | | ==Video Solution (CREATIVE THINKING + ANALYSIS!!!)== |
− | https://youtu.be/WKux87BEO1U | + | https:/90ijn bidxrfgv |
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− | ~Education, the Study of Everything
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− | == Video Solution by OmegaLearn==
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− | https://youtu.be/HISL2-N5NVg?t=817
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− | ~ pi_is_3.14
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− | == Video Solution ==
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− | https://youtu.be/alj9Y8jGNz8
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− | https://youtu.be/meEuDzrM5Ac
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− | ~savannahsolver
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− | ==See Also==
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− | {{AMC8 box|year=2017|num-b=18|num-a=20}}
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− | {{MAA Notice}}
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Latest revision as of 20:38, 10 June 2024