Difference between revisions of "2008 AMC 12B Problems/Problem 23"
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− | == | + | ==Solutions== |
=== Solution 1 === | === Solution 1 === | ||
Every factor of <math>10^n</math> will be of the form <math>2^a \times 5^b , a\leq n , b\leq n</math>. Using the logarithmic property <math>\log(a \times b) = \log(a)+\log(b)</math>, it suffices to count the total number of 2's and 5's running through all possible <math>(a,b)</math>. For every factor <math>2^a \times 5^b</math>, there will be another <math>2^b \times 5^a</math>, 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 <math>\log(2)+\log(5) = \log(10) = 1</math>, the final sum will be the total number of 2's occurring in all factors of <math>10^n</math>. | Every factor of <math>10^n</math> will be of the form <math>2^a \times 5^b , a\leq n , b\leq n</math>. Using the logarithmic property <math>\log(a \times b) = \log(a)+\log(b)</math>, it suffices to count the total number of 2's and 5's running through all possible <math>(a,b)</math>. For every factor <math>2^a \times 5^b</math>, there will be another <math>2^b \times 5^a</math>, 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 <math>\log(2)+\log(5) = \log(10) = 1</math>, the final sum will be the total number of 2's occurring in all factors of <math>10^n</math>. | ||
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=== 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>. |
==See Also== | ==See Also== |
Revision as of 15:38, 5 January 2019
Problem 23
The sum of the base- logarithms of the divisors of is . What is ?
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 .
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 |
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