Mock AIME 1 2007-2008 Problems/Problem 4
Problem
If is an odd number, then find the largest integer that always divides the expression
Solution
Rewrite the expression as Since is odd, let . The expression becomes Consider just the product of the last three terms, , which are consecutive. At least one term must be divisible by and one term must be divisible by then. Also, since there is the term, the expression must be divisible by . Therefore, the minimum integer that always divides the expression must be .
To prove that the number is the largest integer to work, consider when and . These respectively evaluate to be ; their greatest common factor is indeed .
See also
Mock AIME 1 2007-2008 (Problems, Source) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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