MathDB

15

Part of 2005 AMC 10

Problems(2)

Positive Cubes Divide Factorial Expression

Source:

1/12/2009
How many positive integer cubes divide 3!⋅5!⋅7! 3!\cdot 5!\cdot 7!? <spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 6 <span class='latex-bold'>(A)</span>\ 2\qquad <span class='latex-bold'>(B)</span>\ 3\qquad <span class='latex-bold'>(C)</span>\ 4\qquad <span class='latex-bold'>(D)</span>\ 5\qquad <span class='latex-bold'>(E)</span>\ 6
geometry3D geometryfactorialmodular arithmeticAMCAIMEnumber theory
money of &quot;2&quot;s

Source:

11/17/2005
An envelope contains eight bills: 2 2 ones, 2 2 fives, 2 2 tens, and 2 2 twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $20 \$ 20 or more? <spanclass=′latex−bold′>(A)</span> 14<spanclass=′latex−bold′>(B)</span> 27<spanclass=′latex−bold′>(C)</span> 37<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 23 <span class='latex-bold'>(A)</span>\ \frac {1}{4}\qquad <span class='latex-bold'>(B)</span>\ \frac {2}{7}\qquad <span class='latex-bold'>(C)</span>\ \frac {3}{7}\qquad <span class='latex-bold'>(D)</span>\ \frac {1}{2}\qquad <span class='latex-bold'>(E)</span>\ \frac {2}{3}
probability