Difference between revisions of "2005 AMC 10A Problems/Problem 15"
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<math>(3*4)*(3*4)*(2*6)=12^3</math> | <math>(3*4)*(3*4)*(2*6)=12^3</math> | ||
− | So, we have 6 cubes total: <math>1^3 ,2^3, 3^3, 4^3, 6^3,</math> and <math>12^3</math> for a total of <math>6</math> cubes <math>\Rightarrow \mathrm{(E)}</math> | + | So, we have 6 cubes total:<math>1^3 ,2^3, 3^3, 4^3, 6^3,</math> and <math>12^3</math> for a total of <math>6</math> cubes <math>\Rightarrow \mathrm{(E)}</math> |
==See Also== | ==See Also== |
Revision as of 18:55, 10 June 2019
Contents
[hide]Problem
How many positive cubes divide ?
Solution 1
Therefore, a perfect cube that divides must be in the form where , , , and are nonnegative multiples of that are less than or equal to , , and , respectively.
So:
( possibilities)
( possibilities)
( possibility)
( possibility)
So the number of perfect cubes that divide is
Solution 2
In the expression, we notice that there are 3 , 3 , and 3 . This gives us our first 3 cubes: , , and .
However, we can also multiply smaller numbers in the expression to make bigger expressions. For example, (one 2 comes from the , and the other from the ). Using this method, we find:
and
So, we have 6 cubes total: and for a total of cubes
See Also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.