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Difference between revisions of "2000 AMC 8 Problems/Problem 11"
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==Solution==
 
==Solution==
  
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Casework by the units digit <math>u</math> will help organise the answer.
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Casework by the units digit <math>u</math> will help organize the answer.
  
 
<math>u=0</math> gives no solutions, since no real numbers are divisible by <math>0</math>
 
<math>u=0</math> gives no solutions, since no real numbers are divisible by <math>0</math>
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<math>u=3</math> has <math>1</math> solution: <math>33</math>.  <math>\pm 10</math> or <math>\pm 20</math> will retain the units digit, but will stop the number from being divisible by <math>3</math>.  <math>\pm 30</math> is the smallest multiple of <math>10</math> that will keep the number divisible by <math>3</math>, but those numbers are <math>3</math> and <math>63</math>, which are out of the range of the problem.
 
<math>u=3</math> has <math>1</math> solution: <math>33</math>.  <math>\pm 10</math> or <math>\pm 20</math> will retain the units digit, but will stop the number from being divisible by <math>3</math>.  <math>\pm 30</math> is the smallest multiple of <math>10</math> that will keep the number divisible by <math>3</math>, but those numbers are <math>3</math> and <math>63</math>, which are out of the range of the problem.
  
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<math>u=4</math> has <math>2</math> solutions:  <math>24</math> and <math>44</math>.  Adding or subtracting <math>10</math> will kill divisiblty by <math>4</math>, since <math>10</math> is not divisible by <math>4</math>.
+
<math>u=4</math> has <math>2</math> solutions:  <math>24</math> and <math>44</math>.  Adding or subtracting <math>10</math> will kill divisibility by <math>4</math>, since <math>10</math> is not divisible by <math>4</math>.
  
 
<math>u=5</math> has <math>4</math> solutions:  every number ending in <math>5</math> is divisible by <math>5</math>.
 
<math>u=5</math> has <math>4</math> solutions:  every number ending in <math>5</math> is divisible by <math>5</math>.
  
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<math>u=6</math> has <math>1</math> solution:  <math>36</math>.  <math>\pm 10</math> or <math>\pm 20</math> will kill divisibility by <math>3</math>, and thus kill divisiblilty by <math>6</math>.
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<math>u=6</math> has <math>1</math> solution:  <math>36</math>.  <math>\pm 10</math> or <math>\pm 20</math> will kill divisibility by <math>3</math>, and thus kill divisibility by <math>6</math>.
  
 
<math>u=7</math> has no solutions.  The first multiples of <math>7</math> that end in <math>7</math> are <math>7</math> and <math>77</math>, but both are outside of the range of this problem.
 
<math>u=7</math> has no solutions.  The first multiples of <math>7</math> that end in <math>7</math> are <math>7</math> and <math>77</math>, but both are outside of the range of this problem.
  
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<math>u=8</math> has <math>1</math> solution:  <math>48</math>.  <math>\pm 10, \pm 20, \pm 30</math> will all kill divisibility by <math>8</math> since <math>10, 20, </math> and <math>30</math> are not divisibile by <math>8</math>.
+
<math>u=8</math> has <math>1</math> solution:  <math>48</math>.  <math>\pm 10, \pm 20, \pm 30</math> will all kill divisibility by <math>8</math> since <math>10, 20, </math> and <math>30</math> are not divisible by <math>8</math>.
  
 
<math>u=9</math> has no solutions.  <math>9</math> and <math>99</math> are the smallest multiples of <math>9</math> that end in <math>9</math>.
 
<math>u=9</math> has no solutions.  <math>9</math> and <math>99</math> are the smallest multiples of <math>9</math> that end in <math>9</math>.
  
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Totalling the solutions, we have <math>0 + 4 + 4 + 1 + 2 + 4 + 1 + 0 + 1 + 0 = 17</math> solutions, giving the answer <math>\boxed{C}</math>
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Totalling the solutions, we have <math>0 + 4 + 4 + 1 + 2 + 4 + 1 + 0 + 1 + 0 = 17</math> solutions, giving the answer <math>\boxed{C}</math>, which is 17.
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 +
==Video Solution==
 +
https://youtu.be/m5D5-2YB0tI Soo, DRMS, NM
  
 
==See Also==
 
==See Also==

Latest revision as of 14:59, 1 January 2024

Problem

The number $64$ has the property that it is divisible by its unit digit. How many whole numbers between 10 and 50 have this property?

$\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 20$

Solution

Casework by the units digit $u$ will help organize the answer.

$u=0$ gives no solutions, since no real numbers are divisible by $0$

$u=1$ has $4$ solutions, since all numbers are divisible by $1$.

$u=2$ has $4$ solutions, since every number ending in $2$ is even (ie divisible by $2$).

$u=3$ has $1$ solution: $33$. $\pm 10$ or $\pm 20$ will retain the units digit, but will stop the number from being divisible by $3$. $\pm 30$ is the smallest multiple of $10$ that will keep the number divisible by $3$, but those numbers are $3$ and $63$, which are out of the range of the problem.

$u=4$ has $2$ solutions: $24$ and $44$. Adding or subtracting $10$ will kill divisibility by $4$, since $10$ is not divisible by $4$.

$u=5$ has $4$ solutions: every number ending in $5$ is divisible by $5$.

$u=6$ has $1$ solution: $36$. $\pm 10$ or $\pm 20$ will kill divisibility by $3$, and thus kill divisibility by $6$.

$u=7$ has no solutions. The first multiples of $7$ that end in $7$ are $7$ and $77$, but both are outside of the range of this problem.

$u=8$ has $1$ solution: $48$. $\pm 10, \pm 20, \pm 30$ will all kill divisibility by $8$ since $10, 20,$ and $30$ are not divisible by $8$.

$u=9$ has no solutions. $9$ and $99$ are the smallest multiples of $9$ that end in $9$.

Totalling the solutions, we have $0 + 4 + 4 + 1 + 2 + 4 + 1 + 0 + 1 + 0 = 17$ solutions, giving the answer $\boxed{C}$, which is 17.

Video Solution

https://youtu.be/m5D5-2YB0tI Soo, DRMS, NM

See Also

2000 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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
All AJHSME/AMC 8 Problems and Solutions

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