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

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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 $2^a \times 5^b , a\leq n , b\leq n$ . Using the logarithmic property $\log(a \times b) = \log(a)+\log(b)$ , it suffices to count the total number of 2's and 5's running through all possible $(a,b)$ . For every factor $2^a \times 5^b$ , there will be another $2^b \times 5^a$ , 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 $\log(2)+\log(5) = \log(10) = 1$ , the final sum will be the total number of 2's occurring in all factors of $10^n$ .

There are $n+1$ 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 $\log_{10}d_1 + \log_{10}d_2 + \ldots + \log_{10}d_k = 792$ The property $\log(ab) = \log(a)+\log(b)$ now gives $\log_{10}(d_1 d_2\cdot\ldots d_k) = 792$ 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 $d$ 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 $\sum_{p=0}^{n}\sum_{q=0}^{n} \log(2^p5^q) = \sum_{p=0}^{n}\sum_{q=0}^{n}(p\log(2)+q\log(5))$ $= \sum_{p=0}^{n} ((n+1)p\log(2) + \frac{n(n+1)}{2}\log(5))$ $= \frac{n(n+1)^2}{2} \log(2) + \frac{n(n+1)^2}{2} \log(5)$ $= \frac{n(n+1)^2}{2}$ Try for answer choices we get $n=11$

@@ Line 18: / Line 18: @@
 === 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>.
+=== 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==

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
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