Difference between revisions of "Mock AIME 1 2007-2008 Problems/Problem 4"
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Revision as of 16:41, 2 April 2008
Problem
If 
is an odd number, then find the largest integer that always divides the expression
![\[(10x+2)(10x+6)(5x+5)\]](http://latex.artofproblemsolving.com/3/d/8/3d87c25db3731139b72d00b8d2232e15f92cb468.png)
Solution
Rewrite the expression as
![\[4(5x + 1)(5x + 3)(5x+5)\]](http://latex.artofproblemsolving.com/a/1/4/a14042fc131e4f712264a8a8916e4ce0d98e444e.png)
Since is odd, let 
. The expression becomes
![\[4(10n-4)(10n-2)(10n)=32(5n-2)(5n-1)(5n)\]](http://latex.artofproblemsolving.com/5/f/d/5fd22487faa25d82b05aaa1f0fdd6cde5ab18f99.png)
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 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
