Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2005 Alabama ARML TST Problems/Problem 6: Difference between revisions

← Older editNewer edit →
MCrawford (talk | contribs)
1,942 edits
m Exponent change -- it was my error in the solutions packet, though it doesn't change the answer
fixed category
Line 10: Line 10:
*[[2005 Alabama ARML TST Problems/Problem 7 | Next Problem]]
*[[2005 Alabama ARML TST Problems/Problem 7 | Next Problem]]


[[Category:Introductory Number Theory]]
[[Category:Introductory Number Theory Problems]]

Revision as of 19:19, 3 April 2007

Problem

How many of the positive divisors of 3,240,000 are perfect cubes?

Solution

$3240000=2^7\cdot 3^4\cdot 5^4$. We want to know how many numbers are in the form $2^{3a}3^{3b}5^{3c}$ which divide $3,240,000$. This imposes the restrictions $0\leq a\leq 2$,$0 \leq b\leq 1$ and $0 \leq c\leq 1$, which lead to 12 solutions and thus 12 such divisors.

See Also

Retrieved from "https://wiki.aopstest.com/wiki/index.php?title=2005_Alabama_ARML_TST_Problems/Problem_6&oldid=14345"