2010 AMC 12B Problems/Problem 25
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 is incorporated into the giant product.
All numbers , given such that for any integer between and , prime must be less than , contributes a 67 to the product. Considering , the possible values of x are , since are primes that are greater than 67. However, contributes two s to the product, so we must count it twice. Therefore, the answer is .
Similar Solution
After finding the prime factorization of , divide by and add divided by in order to find the total number of multiples of between and . Since ,, and are prime numbers greater than and less than or equal to , subtract from to get the answer .
Need Discussion and Clarification
How do we know that we only have to check 67? There is no solid relationship between 67 being the largest prime factor in 2010 and 67 giving the smallest result of 77. Details in the Discussions page of this Article: http://artofproblemsolving.com/wiki/index.php?title=Talk:2010_AMC_12B_Problems/Problem_25
See Also
2010 AMC 12B (Problems • Answer Key • Resources) | |
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