Difference between revisions of "2016 AMC 12B Problems/Problem 22"

Latest revision as of 23:33, 9 November 2024

Problem

For a certain positive integer $n$ less than $1000$ , the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$ , a repeating decimal of period of $6$ , and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$ , a repeating decimal of period $4$ . In which interval does $n$ lie?

$\textbf{(A)}\ [1,200]\qquad\textbf{(B)}\ [201,400]\qquad\textbf{(C)}\ [401,600]\qquad\textbf{(D)}\ [601,800]\qquad\textbf{(E)}\ [801,999]$

Solution

Solution by e_power_pi_times_i

If $\frac{1}{n} = 0.\overline{abcdef}$ , $n$ must be a factor of $999999$ . Also, by the same procedure, $n+6$ must be a factor of $9999$ . Checking through all the factors of $999999$ and $9999$ that are less than $1000$ , we see that $n = 297$ is a solution, so the answer is $\boxed{\textbf{(B)}}$ .

Note: $n = 27$ and $n = 3$ are both solutions, which invalidates this method. However, we need to examine all factors of $999999$ that are not factors of $99999$ , $999$ , or $99$ , or $9$ . Additionally, we need $n+6$ to be a factor of $9999$ but not $999$ , $99$ , or $9$ . Indeed, $297$ satisfies these requirements.

We can see that $n=27$ and $n=3$ are not solutions by checking it in the requirements of the problem: $\frac{1}{3}=0.3333\dots$ , period 1, and $\frac{1}{27}=0.037037\dots$ , period 3. Thus, $n=297$ is the only answer.

For anyone who wants more information about repeating decimals, visit: https://en.wikipedia.org/wiki/Repeating_decimal

Solution 2 (Faster Approach)

Notice that the repeating fraction $0.\overline{abcdef}$ can be represented as $\frac{abcdef}{999999},$ and thereby, $n|999999.$ Also, notice that $0.\overline{wxyz} = \frac{wxyz}{9999},$ so $(n+6)|9999.$ However, we have to make some restrictions here. For instance, if $n|99999,$ then $\frac{1}{n}$ could be expressed as $\frac{a’b’c’d’e’}{99999} = .\overline{a’b’c’d’e’}$ which cannot happen. Therefore, from this, we see that the smallest $m$ such that $n|\underbrace{999\cdots999}_{m \text{ nines}}$ is $m = 6.$ Also, the smallest number $m$ such that $(n+6)|\underbrace{999\cdots999}_{m \text{ nines}}$ is $m = 4$ by similar reasoning.

Proceeding, we can factorize $9999 = 99 \times 101,$ after which we see that $n+6$ must contain a prime factor of $101$ as it cannot divide $99$ but must divide $9999.$ However, $101$ is prime, so $101|(n+6)$ ! Looking at the answer choices, all of the intervals are less than $1000,$ so we know that (the minimum value of) $n+6$ is thereby either $101, 101 \times 3,$ or $101 \times 9.$ Testing, we see that $n+6 = 303$ gives $n = 297 = 3^3 \times 11,$ which in fact is a divisor of $999 \times 1001$ while not being a divisor of $999.$ Therefore, the answer is $\boxed{\text{(B)}}.$

~ Professor-Mom

Video Solution by CanadaMath (Problem 21-25)

Fast Forward to 11:20 for problem 22 https://www.youtube.com/watch?v=P3jJDLGyF2w&t=1546s

~THEMATHCANADIAN

@@ Line 1: / Line 1: @@
-=Problem=
+==Problem==
-For a certain positive integer n less than <math>1000</math>, the decimal equivalent of <math>\frac{1}{n}</math> is <math>0.\overline{abcdef}</math>, a repeating decimal of period of 6, and the decimal equivalent of <math>\frac{1}{n+6}</math> is <math>0.\overline{wxyz}</math>, a repeating decimal of period 4. In which interval does <math>n</math> lie?
+For a certain positive integer <math>n</math> less than <math>1000</math>, the decimal equivalent of <math>\frac{1}{n}</math> is <math>0.\overline{abcdef}</math>, a repeating decimal of period of <math>6</math>, and the decimal equivalent of <math>\frac{1}{n+6}</math> is <math>0.\overline{wxyz}</math>, a repeating decimal of period <math>4</math>. In which interval does <math>n</math> lie?
-<math>\textbf{(A)}\ [1,200]\qquad\textbf{(B)}\ [201,400]\qquad\textbf{(C)}\ [401,600]1\qquad\textbf{(D)}\ [601,800]\qquad\textbf{(E)}\ [801,999]</math>
+<math>\textbf{(A)}\ [1,200]\qquad\textbf{(B)}\ [201,400]\qquad\textbf{(C)}\ [401,600]\qquad\textbf{(D)}\ [601,800]\qquad\textbf{(E)}\ [801,999]</math>
+[[Category: Intermediate Number Theory Problems]]
+==Solution==
+Solution by e_power_pi_times_i
+If <math>\frac{1}{n} = 0.\overline{abcdef}</math>, <math>n</math> must be a factor of <math>999999</math>. Also, by the same procedure, <math>n+6</math> must be a factor of <math>9999</math>. Checking through all the factors of <math>999999</math> and <math>9999</math> that are less than <math>1000</math>, we see that <math>n = 297</math> is a solution, so the answer is <math>\boxed{\textbf{(B)}}</math>.
+Note: <math>n = 27</math> and <math>n = 3</math> are both solutions, which invalidates this method. However, we need to examine all factors of <math>999999</math> that are not factors of <math>99999</math>, <math>999</math>, or <math>99</math>, or <math>9</math>. Additionally, we need <math>n+6</math> to be a factor of <math>9999</math> but not <math>999</math>, <math>99</math>, or <math>9</math>. Indeed, <math>297</math> satisfies these requirements.
+We can see that <math>n=27</math> and <math>n=3</math> are not solutions by checking it in the requirements of the problem: <math>\frac{1}{3}=0.3333\dots</math>, period 1, and <math>\frac{1}{27}=0.037037\dots</math>, period 3. Thus, <math>n=297</math> is the only answer.
+For anyone who wants more information about repeating decimals, visit: https://en.wikipedia.org/wiki/Repeating_decimal
+==Solution 2 (Faster Approach)==
+Notice that the repeating fraction <math>0.\overline{abcdef}</math> can be represented as <math>\frac{abcdef}{999999},</math> and thereby, <math>n|999999.</math> Also, notice that <math>0.\overline{wxyz} = \frac{wxyz}{9999},</math> so <math>(n+6)|9999.</math> However, we have to make some restrictions here. For instance, if <math>n|99999,</math> then <math>\frac{1}{n}</math> could be expressed as <math>\frac{a’b’c’d’e’}{99999} = .\overline{a’b’c’d’e’}</math> which cannot happen. Therefore, from this, we see that the smallest <math>m</math> such that <math>n|\underbrace{999\cdots999}_{m \text{ nines}}</math> is <math>m = 6.</math> Also, the smallest number <math>m</math> such that <math>(n+6)|\underbrace{999\cdots999}_{m \text{ nines}}</math> is <math>m = 4</math> by similar reasoning.
+Proceeding, we can factorize <math>9999 = 99 \times 101,</math> after which we see that <math>n+6</math> must contain a prime factor of <math>101</math> as it cannot divide <math>99</math> but must divide <math>9999.</math> However, <math>101</math> is prime, so <math>101|(n+6)</math>! Looking at the answer choices, all of the intervals are less than <math>1000,</math> so we know that (the minimum value of) <math>n+6</math> is thereby either <math>101, 101 \times 3,</math> or <math>101 \times 9.</math> Testing, we see that <math>n+6 = 303</math> gives <math>n = 297 = 3^3 \times 11,</math> which in fact is a divisor of <math>999 \times 1001</math> while not being a divisor of <math>999.</math> Therefore, the answer is <math>\boxed{\text{(B)}}.</math>
+~ Professor-Mom
+==Video Solution by CanadaMath (Problem 21-25)==
+Fast Forward to 11:20 for problem 22
+https://www.youtube.com/watch?v=P3jJDLGyF2w&t=1546s
+~THEMATHCANADIAN
+==See Also==
+{{AMC12 box|year=2016|ab=B|num-b=21|num-a=23}}
+{{MAA Notice}}

2016 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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