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

Revision as of 12:47, 15 February 2021

Problem

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 $2^{n-a} \times 5^{n-b}$ . Note this is not true if $10^n$ is a perfect square. When these are added, they equal $2^{a+n-a} \times 5^{b+n-b} = 10^n$ . $\log 10^n=n$ so the number of factors divided by 2 times n equals the sum of all the factors, 792.

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 $(n+1)^2$ total factors. $\frac{(n+1)^2}{2n} = 792$ . We then plug in answer choices and arrive at the answer $\boxed {11}$

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 $\sum_{p=0}^{n}\sum_{q=0}^{n} \log(2^p5^q) = \sum_{p=0}^{n}\sum_{q=0}^{n}(p\log(2)+q\log(5))$ $= \sum_{p=0}^{n} ((n+1)p\log(2) + \frac{n(n+1)}{2}\log(5))$ $= \frac{n(n+1)^2}{2} \log(2) + \frac{n(n+1)^2}{2} \log(5)$ $= \frac{n(n+1)^2}{2}$ Trying for answer choices we get $n=11$

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 $11$ .

Revision as of 21:55, 18 January 2021 (view source) Eagleye (talk \| contribs) (→‎Solution 1) ← Older edit		Revision as of 12:47, 15 February 2021 (view source) Etvat (talk \| contribs) Newer edit →
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−	== Problem 23 ==	+	== Problem ==
	The sum of the base-<math>10</math> logarithms of the divisors of <math>10^n</math> is <math>792</math>. What is <math>n</math>?		The sum of the base-<math>10</math> logarithms of the divisors of <math>10^n</math> is <math>792</math>. What is <math>n</math>?

2008 AMC 12B (Problems • Answer Key • Resources)
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