Difference between revisions of "2005 AMC 10A Problems/Problem 15"
Dairyqueenxd (talk | contribs) (→Solution 1) |
Dairyqueenxd (talk | contribs) (→Solution 2) |
||
Line 34: | Line 34: | ||
<math>(3 \cdot 4) \cdot (3 \cdot 4) \cdot (2 \cdot 6) = 12^3</math> | <math>(3 \cdot 4) \cdot (3 \cdot 4) \cdot (2 \cdot 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 | + | 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>\boxed{\textbf{(E) }6}</math> cubes |
==See also== | ==See also== |
Latest revision as of 18:29, 13 December 2021
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
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.