Difference between revisions of "2008 AMC 12B Problems/Problem 23"
Mutinykids (talk | contribs) m (→Solution) |
(→Solution 3) |
||
Line 18: | Line 18: | ||
=== Solution 3 === | === Solution 3 === | ||
For every divisor <math>d</math> of <math>10^n</math>, <math>d \le \sqrt{10^n}</math>, we have <math>\log d + \log \frac{10^n}{d} = \log 10^n = n</math>. There are <math>\left \lfloor \frac{(n+1)^2}{2} \right \rfloor</math> divisors of <math>10^n = 2^n \times 5^n</math> that are <math>\le \sqrt{10^n}</math>. After casework on the parity of <math>n</math>, we find that the answer is given by <math>n \times \frac{(n+1)^2}{2} = 792 \Longrightarrow n = 11\ \mathrm{(A)}</math>. | For every divisor <math>d</math> of <math>10^n</math>, <math>d \le \sqrt{10^n}</math>, we have <math>\log d + \log \frac{10^n}{d} = \log 10^n = n</math>. There are <math>\left \lfloor \frac{(n+1)^2}{2} \right \rfloor</math> divisors of <math>10^n = 2^n \times 5^n</math> that are <math>\le \sqrt{10^n}</math>. After casework on the parity of <math>n</math>, we find that the answer is given by <math>n \times \frac{(n+1)^2}{2} = 792 \Longrightarrow n = 11\ \mathrm{(A)}</math>. | ||
+ | |||
+ | === Solution 4 === | ||
+ | The sum is | ||
+ | <cmath> \sum_{p=0}^{n}\sum_{q=0}^{n} \log(2^p5^q) = \sum_{p=0}^{n}\sum_{q=0}^{n}(p\log(2)+q\log(5)) </cmath> | ||
+ | <cmath> = \sum_{p=0}^{n} ((n+1)p\log(2) + \frac{n(n+1)}{2}\log(5)) </cmath> | ||
+ | <cmath>= \frac{n(n+1)^2}{2} \log(2) + \frac{n(n+1)^2}{2} \log(5) </cmath> | ||
+ | <cmath> = \frac{n(n+1)^2}{2} </cmath> | ||
+ | Try for answer choices we get <math>n=11</math> | ||
==See Also== | ==See Also== |
Revision as of 10:26, 11 January 2019
Problem 23
The sum of the base- logarithms of the divisors of
is
. What is
?
Contents
Solutions
Solution 1
Every factor of will be of the form
. Using the logarithmic property
, it suffices to count the total number of 2's and 5's running through all possible
. For every factor
, there will be another
, so it suffices to count the total number of 2's occurring in all factors (because of this symmetry, the number of 5's will be equal). And since
, the final sum will be the total number of 2's occurring in all factors of
.
There are choices for the exponent of 5 in each factor, and for each of those choices, there are
factors (each corresponding to a different exponent of 2), yielding
total 2's. The total number of 2's is therefore
. Plugging in our answer choices into this formula yields 11 (answer choice
) as the correct answer.
Solution 2
We are given The property
now gives
The product of the divisors is (from elementary number theory)
where
is the number of divisors. Note that
, so
. Substituting these values with
in our equation above, we get
, from whence we immediately obtain
as the correct answer.
Solution 3
For every divisor of
,
, we have
. There are
divisors of
that are
. After casework on the parity of
, we find that the answer is given by
.
Solution 4
The sum is
Try for answer choices we get
See Also
2008 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.