,
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
satisfying
![\[
\text{Find all non-negative integers } (a, b, c, n) \text{ such that}
\]](http://latex.artofproblemsolving.com/9/5/9/9591f7c842d1b6afcba11b15b3ea3f9e01bc937a.png)
![\[
2^a + 3^b + 5^c = n!
\]](http://latex.artofproblemsolving.com/b/c/6/bc6b0754dd874f646fcab124c91583b620db273c.png)
Y by Adventure10
Let 
and 
be integers with 
. For a positive integer 
, let 
be the least common multiple of 
.
Prove that is a divisor of ![$L_i \cdot [\binom{m}{i} - \binom{m-k}{i}]$](//latex.artofproblemsolving.com/9/f/3/9f39bfc257d726b7c009aeb2e17bc8a2c1bfad08.png)
for all 
. [Here, 
denotes a binomial coefficient. Note that 
if 
.]
and 
be integers with 
. For a positive integer 
, let 
be the least common multiple of 
.Prove that
is a divisor of ![$L_i \cdot [\binom{m}{i} - \binom{m-k}{i}]$](http://latex.artofproblemsolving.com/9/f/3/9f39bfc257d726b7c009aeb2e17bc8a2c1bfad08.png)
for all 
. [Here, 
denotes a binomial coefficient. Note that 
if 
.]
