Question 1009054
If the product represented by 274! Is divisible by 12 to the power 
of n, what is the largest possible value of n?
<pre>
n will be the number of factors of 12 there are in 274!

12 = 2×2×3

Each factor of 12 in 274! amounts to two factors of 2 and 
one factor of 3.

We are interested in how many factors of 2 and how many 
factors of 3 are contained in

a)   274! = 1×2×3×4×...×274

Let's first find out how many factors of 3 are contained in 274!

Product a) contains 

 91 multiples of 3, since 274/3 = 91.333...
 30 multiples of 3^2, or 9, since 274/9 = 30.444...
 10 multiples of 3^3, or 27, since 274/27 = 10.148...
  3 multiples of 3^4, or 81, since 274/81 = 3.3827... 
  1 multiple of 3^5, or 243, since 274/243 = 1.12767...
--------------------------
135 factors of 3 contained in 274!

Since every factor of 12 contained in 274! has exactly 1
factor of 3, 135 will be the number of factors of 12 in 
274!, provided that there are at least twice that many
factors of 2 in 274!, since each factor of 12 amounts to 
2 factors of 2 and 1 factor of 12.  So we must make sure 
that 274! contains at least twice 135 or 270 factors of 
2 in order to claim that it has 135 factors of 12.  So 
let's find out if there are enough factors of 2 to 
justify that 135 is the correct answer.

a)   274! = 1×2×3×4×...×274

Product a) contains 

137 multiples of 2, since 274/2 = 137
 68 multiples of 2^2, or 4, since 274/4 = 68.5
 34 multiples of 2^3, or 8, since 274/8 = 34.25
 17 multiples of 2^4, or 16, since 274/16 = 17.125 
  8 multiples of 2^5, or 32, since 274/32 = 8.5625
  4 multiples of 2^6, or 64, since 274/64 = 4.28125
  2 multiples of 2^7, or 128, since 274/128 = 2.140625
  1 multiple of 2^8, or 256, since 274/256 = 1.0703125
----------------------------
271 factors of 2 contained in 274!

So there is just 1 extra factor of 2 than 270, the number 
necessary to make there be 135 factors of 12 in 274!

Answer: 274! contains 135 factors of 12 Therefore n = 135 
is the largest possible value of n. 

Edwin</pre>