Difference between revisions of "2010 AMC 12B Problems/Problem 25"
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<math>\textbf{(A)}\ 74 \qquad \textbf{(B)}\ 75 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 77 \qquad \textbf{(E)}\ 78</math> | <math>\textbf{(A)}\ 74 \qquad \textbf{(B)}\ 75 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 77 \qquad \textbf{(E)}\ 78</math> | ||
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== Solution == | == Solution == | ||
Revision as of 23:00, 31 January 2012
Problem 25
For every integer 
, let 
be the largest power of the largest prime that divides 
. For example 
. What is the largest integer 
such that 
divides
?
Solution
Because 67 is the largest prime factor of 2010, it means that in the prime factorization of , there'll be 
where 
is the desired value we are looking for. Thus, to find this answer, we need to look for the number of times 67 would be incorporated into the giant product.
Any number of the form 
would fit this form. However, this number tops at 
because 71 is a higher prime than 67. 
itself must be counted twice because it's counted twice as a squared number. Any non-prime number that's less than 79 (and greater than 71) can be counted, and this totals 5. Thus, 
See Also
| 2010 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 24 |
Followed by Last Problem |
| 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 AMC 12 Problems and Solutions | |
