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Difference between revisions of "2008 AMC 12B Problems/Problem 23"

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== Solutions ==
 
== Solutions ==
 
=== Solution 1 ===
 
=== Solution 1 ===
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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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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>. Not all of these base ten logarithms will be rational, but we can add them together in a certain way to make it rational. Recall the logarithmic property <math>\log(a \times b) = \log(a)+\log(b)</math>. For any factor <math>2^a \times 5^b</math>, there will be another factor <math>2^n-a \times 5^n-b</math>.(This is not true if 10^n is a perfect square.) When these are added, they equal 2^a+n-a \times 5^b+n-b<math> = 10^n. Log 10^n=n. This means the number of factors divided by 2 times n equals the sum of all the factors, 792.  
−
 
+
−
There are <math>n+1</math> choices for the exponent of 5 in each factor, and for each of those choices, there are <math>n+1</math> factors (each corresponding to a different exponent of 2), yielding <math>0+1+2+3...+n = \frac{n(n+1)}{2}</math> total 2's. The total number of 2's is therefore <math>\frac{n \cdot(n+1)^2}{2} = \frac{n^3+2n^2+n}{2}</math>. Plugging in our answer choices into this formula yields 11 (answer choice <math>\mathrm{(A)}</math>) as the correct answer.
+
There are </math>n+1<math> choices for the exponent of 5 in each factor, and for each of those choices, there are </math>n+1<math> factors (each corresponding to a different exponent of 2), yielding </math>0+1+2+3...+n = \frac{n(n+1)}{2}<math> total 2's. The total number of 2's is therefore </math>\frac{n \cdot(n+1)^2}{2} = \frac{n^3+2n^2+n}{2}<math>. Plugging in our answer choices into this formula yields 11 (answer choice </math>\mathrm{(A)}<math>) as the correct answer.
  
 
=== Solution 2 ===
 
=== Solution 2 ===
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We are given <cmath> \log_{10}d_1 + \log_{10}d_2 + \ldots + \log_{10}d_k = 792 </cmath> The property <math>\log(ab) = \log(a)+\log(b)</math> now gives <cmath> \log_{10}(d_1 d_2\cdot\ldots d_k) = 792 </cmath> The product of the divisors is (from elementary number theory) <math>a^{d(n)/2}</math> where <math>d(n)</math> is the number of divisors. Note that <math>10^n = 2^n\cdot 5^n</math>, so <math>d(n) = (n + 1)^2</math>. Substituting these values with <math>a = 10^n</math> in our equation above, we get <math>n(n + 1)^2 = 1584</math>, from whence we immediately obtain <math>\framebox[1.2\width]{(A)}</math> as the correct answer.
+
We are given <cmath> \log_{10}d_1 + \log_{10}d_2 + \ldots + \log_{10}d_k = 792 </cmath> The property </math>\log(ab) = \log(a)+\log(b)<math> now gives <cmath> \log_{10}(d_1 d_2\cdot\ldots d_k) = 792 </cmath> The product of the divisors is (from elementary number theory) </math>a^{d(n)/2}<math> where </math>d(n)<math> is the number of divisors. Note that </math>10^n = 2^n\cdot 5^n<math>, so </math>d(n) = (n + 1)^2<math>. Substituting these values with </math>a = 10^n<math> in our equation above, we get </math>n(n + 1)^2 = 1584<math>, from whence we immediately obtain </math>\framebox[1.2\width]{(A)}<math> as the correct answer.
  
 
=== 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 ===
 
=== Solution 4 ===
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<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} \log(2) + \frac{n(n+1)^2}{2} \log(5) </cmath>
 
<cmath> = \frac{n(n+1)^2}{2} </cmath>
 
<cmath> = \frac{n(n+1)^2}{2} </cmath>
−
Trying for answer choices we get <math>n=11</math>
+
Trying for answer choices we get </math>n=11$
  
 
== Alternative thinking ==
 
== Alternative thinking ==

Revision as of 08:01, 14 December 2020

Problem 23

The sum of the base-$10$ logarithms of the divisors of $10^n$ is $792$. What is $n$?

$\text{(A)}\ 11\qquad \text{(B)}\ 12\qquad \text{(C)}\ 13\qquad \text{(D)}\ 14\qquad \text{(E)}\ 15$

Solutions

Solution 1

Every factor of $10^n$ will be of the form $2^a \times 5^b , a\leq n , b\leq n$. Not all of these base ten logarithms will be rational, but we can add them together in a certain way to make it rational. Recall the logarithmic property $\log(a \times b) = \log(a)+\log(b)$. For any factor $2^a \times 5^b$, there will be another factor $2^n-a \times 5^n-b$.(This is not true if 10^n is a perfect square.) When these are added, they equal 2^a+n-a \times 5^b+n-b$= 10^n. Log 10^n=n. This means the number of factors divided by 2 times n equals the sum of all the factors, 792.

There are$ (Error compiling LaTeX. Unknown error_msg)n+1$choices for the exponent of 5 in each factor, and for each of those choices, there are$n+1$factors (each corresponding to a different exponent of 2), yielding$0+1+2+3...+n = \frac{n(n+1)}{2}$total 2's. The total number of 2's is therefore$\frac{n \cdot(n+1)^2}{2} = \frac{n^3+2n^2+n}{2}$. Plugging in our answer choices into this formula yields 11 (answer choice$\mathrm{(A)}$) as the correct answer.

=== Solution 2 === We are given <cmath> \log_{10}d_1 + \log_{10}d_2 + \ldots + \log_{10}d_k = 792 </cmath> The property$ (Error compiling LaTeX. Unknown error_msg)\log(ab) = \log(a)+\log(b)$now gives <cmath> \log_{10}(d_1 d_2\cdot\ldots d_k) = 792 </cmath> The product of the divisors is (from elementary number theory)$a^{d(n)/2}$where$d(n)$is the number of divisors. Note that$10^n = 2^n\cdot 5^n$, so$d(n) = (n + 1)^2$. Substituting these values with$a = 10^n$in our equation above, we get$n(n + 1)^2 = 1584$, from whence we immediately obtain$\framebox[1.2\width]{(A)}$as the correct answer.

=== Solution 3 === For every divisor$ (Error compiling LaTeX. Unknown error_msg)d$of$10^n$,$d \le \sqrt{10^n}$, we have$\log d + \log \frac{10^n}{d} = \log 10^n = n$. There are$\left \lfloor \frac{(n+1)^2}{2} \right \rfloor$divisors of$10^n = 2^n \times 5^n$that are$\le \sqrt{10^n}$. After casework on the parity of$n$, we find that the answer is given by$n \times \frac{(n+1)^2}{2} = 792 \Longrightarrow n = 11\ \mathrm{(A)}$.

=== 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> Trying for answer choices we get$ (Error compiling LaTeX. Unknown error_msg)n=11$

Alternative thinking

After arriving at the equation $n(n+1)^2 = 1584$, notice that all of the answer choices are in the form $10+k$, where $k$ is $1-5$. We notice that the ones digit of $n(n+1)^2$ is $4$, and it is dependent on the ones digit of the answer choices. Trying $1-5$ for $n$, we see that only $1$ yields a ones digit of $4$, so our answer is $11$.

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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