If the product represented by 274! Is divisible by 12 to the power
of n, what is the largest possible value of n?
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...
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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
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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