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

Latest revision as of 17:02, 17 January 2023

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 multiplied, 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. $\frac{(n+1)^2}{2}*n = 792$ . 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 $n=11$

Solution 5

Let integer $d$ be the number of divisors $10^n$ has. Then, we set up $\frac{d}{2}$ pairs of divisors such that each pair $(a,b)$ satisfies $ab = 10^n$ -- ex. $(1, 10^n), (2, 5*10^{n-1})$ , etc. Then the sum of the base- $10$ logarithms is $\sum_{j=1}^{\frac{d}{2}} \log_{10} a_j + \log_{10} b_j = \sum_{j=1}^{\frac{d}{2}} \log_{10} a_j b_j = \sum_{j=1}^{\frac{d}{2}} \log_{10} 10^n = \sum_{j=1}^{\frac{d}{2}} n = \frac{nd}{2}$ We can use the property that only perfect squares have an odd number of factors, as for perfect square $p^2$ , we have ordered pair $(p,p)$ that works. For even $n$ , then, $10^{\frac{n}{2}}$ can be multiplied by itself to get $10^n$ , so $d$ is odd. But, in our summation, $\frac{d}{2}$ does not exist for even $n$ as $d$ is then odd, so $d$ must be even. And, since $\frac{nd}{2}$ = $792$ , We want to find a $d$ for our odd options $n$ such that $nd$ = $1584$ . For $n=11$ , $d=144$ works, and integer $d$ can not be found for other odd $n$ . So, we get $\framebox[1.2\width]{(A) 11}$

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 $11$ .

@@ Line 1: / Line 1: @@
-== 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>?
@@ Line 6: / Line 6: @@
 == Solutions ==
 === 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>. 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 </math>5^(n-b)<math>.(This is not true if </math>10^n<math> is a perfect square.) When these are added, they equal 2^(a+n-a)</math> \times 5^(b+n-b)<math> = </math>10^n. Log <math>10^n=n. This means the number of factors divided by 2 times n equals the sum of all the factors, 792.
+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>. Note this is not true if <math>10^n</math> is a perfect square. When these are multiplied, they equal <math>2^{a+n-a} \times 5^{b+n-b} = 10^n</math>. <math>\log 10^n=n</math> so 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 factors. So we have \frac{n(n+1)}{2}</math> divided by 2 times n = 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>(n+1)^2</math> total factors. <math>\frac{(n+1)^2}{2}*n = 792</math>. We then plug in answer choices and arrive at the answer <math>\boxed {11}</math>
 === Solution 2 ===
@@ Line 23: / Line 23: @@
 <cmath> = \frac{n(n+1)^2}{2} </cmath>
 Trying for answer choices we get <math>n=11</math>
+== Solution 5 ==
+Let integer <math>d</math> be the number of divisors <math>10^n</math> has. Then, we set up <math>\frac{d}{2}</math> pairs of divisors such that each pair <math>(a,b)</math> satisfies <math>ab = 10^n</math> -- ex. <math>(1, 10^n), (2, 5*10^{n-1})</math>, etc. Then the sum of the base-<math>10</math> logarithms is <math>\sum_{j=1}^{\frac{d}{2}} \log_{10} a_j + \log_{10} b_j = \sum_{j=1}^{\frac{d}{2}} \log_{10} a_j b_j = \sum_{j=1}^{\frac{d}{2}} \log_{10} 10^n = \sum_{j=1}^{\frac{d}{2}} n = \frac{nd}{2}</math>
+We can use the property that only perfect squares have an odd number of factors, as for perfect square <math>p^2</math>, we have ordered pair <math>(p,p)</math> that works. For even <math>n</math>, then, <math>10^{\frac{n}{2}}</math> can be multiplied by itself to get <math>10^n</math>, so <math>d</math> is odd. But, in our summation, <math>\frac{d}{2}</math> does not exist for even <math>n</math> as <math>d</math> is then odd, so <math>d</math> must be even. And, since <math>\frac{nd}{2}</math> = <math>792</math>, We want to find a <math>d</math> for our odd options <math>n</math> such that <math>nd</math> = <math>1584</math>. For <math>n=11</math>, <math>d=144</math> works, and integer <math>d</math> can not be found for other odd <math>n</math>. So, we get <math>\framebox[1.2\width]{(A) 11}</math>
 == Alternative thinking ==

2008 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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