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Difference between revisions of "2020 AMC 8 Problems/Problem 19"

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==Problem 19==
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==Problem==
 
A number is called flippy if its digits alternate between two distinct digits. For example, <math>2020</math> and <math>37373</math> are flippy, but <math>3883</math> and <math>123123</math> are not. How many five-digit flippy numbers are divisible by <math>15?</math>
 
A number is called flippy if its digits alternate between two distinct digits. For example, <math>2020</math> and <math>37373</math> are flippy, but <math>3883</math> and <math>123123</math> are not. How many five-digit flippy numbers are divisible by <math>15?</math>
  
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==Solution 1==
 
==Solution 1==
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A number is divisible by <math>15</math> precisely if it is divisible by <math>3</math> and <math>5</math>. The latter means the last digit must be either <math>5</math> or <math>0</math>, and the former means the sum of the digits must be divisible by <math>3</math>. If the last digit is <math>0</math>, the first digit would be <math>0</math> (because the digits alternate), which is not possible. Hence the last digit must be <math>5</math>, and the number is of the form <math>5\square 5\square 5</math>. If the unknown digit is <math>x</math>, we deduce <math>5+x+5+x+5 \equiv 0 \pmod{3} \Rightarrow 2x \equiv 0 \pmod{3}</math>. We know <math>2^{-1}</math> exists modulo <math>3</math> because 2 is relatively prime to 3, so we may conclude that <math>x</math> (i.e. the second and fourth digit of the number) must be a multiple of <math>3</math>. It can be <math>0</math>, <math>3</math>, <math>6</math>, or <math>9</math>, so there are <math>\boxed{\textbf{(B) }4}</math> options: <math>50505</math>, <math>53535</math>, <math>56565</math>, and <math>59595</math>.
  
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To be divisible by <math>15</math>, a number must first be divisible by <math>3</math> and <math>5</math>. By divisibility rules, the last digit must be either <math>5</math> or <math>0</math>, and the sum of the digits must be divisible by <math>3</math>. If the last digit is <math>0</math>, the first digit would be <math>0</math> (because the digits alternate). So, the last digit must be <math>5</math>, and we have <cmath>5+x+5+x+5 \equiv 0 \pmod{3}</cmath> <cmath>2x \equiv 0 \pmod{3}</cmath> We know the inverse exists because 2 is relatively prime to 3, and thus we can conclude that <math>x</math> (or the second and fourth digits) is always a multiple of <math>3</math>. We have 4 options: <math>0, 3, 6, 9</math>, and our answer is <math>4</math> and <math>\boxed{\textbf{(B) } 4}</math>    ~samrocksnature
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==Solution 2 (variant of Solution 1)==
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As in Solution 1, we find that such numbers must start with <math>5</math> and alternate with <math>5</math> (i.e. must be of the form <math>5\square 5\square 5</math>), where the two digits between the <math>5</math>s need to be the same. Call that digit <math>x</math>. For the number to be divisible by <math>3</math>, the sum of the digits must be divisible by <math>3</math>; since the sum of the three <math>5</math>s is <math>15</math>, which is already a multiple of <math>3</math>, it must also be the case that <math>x+x=2x</math> is a multiple of <math>3</math>. Thus, the problem reduces to finding the number of digits from <math>0</math> to <math>9</math> for which <math>2x</math> is a multiple of <math>3</math>. This leads to <math>x=0</math>, <math>3</math>, <math>6</math>, or <math>9</math>, so there are <math>\boxed{\textbf{(B) }4}</math> possible numbers (namely <math>50505</math>, <math>53535</math>, <math>56565</math>, and <math>59595</math>).
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==Solution 2==
 
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A number that is divisible by <math>15</math> must be divisible by <math>3</math> and <math>5</math>. To be divisible by <math>3</math>, the sum of its digits must be divisible by <math>3</math> and to be divisible by <math>5</math>, it must end in a <math>5</math> or a <math>0</math>. Observe that a five-digit flippy number must start and end with the same digit. Since a five-digit number cannot start with <math>0</math>, it also cannot end in <math>0</math>. This means that the numbers that we are looking for must end in <math>5</math>. This also means that they must start with <math>5</math> and alternate with <math>5</math> (i.e. the number must be of the form <math>5\square5\square5</math>). The two digits between the <math>5s</math> must be the same. Let's call that digit <math>x</math>. We know that the sum of the digits must be a multiple of <math>3</math>. Since the sum of the three <math>5s</math> is <math>15</math> which is already a multiple of <math>3</math>, for the entire five-digit number to be a multiple of <math>3</math>, it must also be the case that <math>x+x=2x</math> is also a multiple of <math>3</math>. Thus, the problem reduces to finding the number of digits from <math>0</math> to <math>9</math> for which <math>2x</math> is a multiple of <math>3</math>. This leads to <math>x=0,3,6,</math> and <math>9</math> and we have four valid answers (i.e. <math>50505,53535,56565,</math> and <math>59595</math>) <math>\implies\boxed{\textbf{(B) } 4}</math>.<br>
 
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~[http://artofproblemsolving.com/community/user/jmansuri junaidmansuri]
 
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==Solution 3==
 
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A flippy number is of the form <math>5X5X5</math> where <math>x\in\{0,3,6,9\}</math> by the divisibility rules for <math>3</math> and <math>5</math> so the answer is <math>\textbf{(B) }4</math>.
 
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-franzliszt
 
  
 
==See also==  
 
==See also==  
 
{{AMC8 box|year=2020|num-b=18|num-a=20}}
 
{{AMC8 box|year=2020|num-b=18|num-a=20}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 08:50, 20 November 2020

Problem

A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$

$\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8$

Solution 1

A number is divisible by $15$ precisely if it is divisible by $3$ and $5$. The latter means the last digit must be either $5$ or $0$, and the former means the sum of the digits must be divisible by $3$. If the last digit is $0$, the first digit would be $0$ (because the digits alternate), which is not possible. Hence the last digit must be $5$, and the number is of the form $5\square 5\square 5$. If the unknown digit is $x$, we deduce $5+x+5+x+5 \equiv 0 \pmod{3} \Rightarrow 2x \equiv 0 \pmod{3}$. We know $2^{-1}$ exists modulo $3$ because 2 is relatively prime to 3, so we may conclude that $x$ (i.e. the second and fourth digit of the number) must be a multiple of $3$. It can be $0$, $3$, $6$, or $9$, so there are $\boxed{\textbf{(B) }4}$ options: $50505$, $53535$, $56565$, and $59595$.

Solution 2 (variant of Solution 1)

As in Solution 1, we find that such numbers must start with $5$ and alternate with $5$ (i.e. must be of the form $5\square 5\square 5$), where the two digits between the $5$s need to be the same. Call that digit $x$. For the number to be divisible by $3$, the sum of the digits must be divisible by $3$; since the sum of the three $5$s is $15$, which is already a multiple of $3$, it must also be the case that $x+x=2x$ is a multiple of $3$. Thus, the problem reduces to finding the number of digits from $0$ to $9$ for which $2x$ is a multiple of $3$. This leads to $x=0$, $3$, $6$, or $9$, so there are $\boxed{\textbf{(B) }4}$ possible numbers (namely $50505$, $53535$, $56565$, and $59595$).

See also

2020 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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All AJHSME/AMC 8 Problems and Solutions

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